What deuterocanonical books are quoted in the New Testament?

• Matt. 6:19-20 - Jesus' statement about laying up for yourselves treasure in heaven follows Sirach 29:11 - lay up your treasure.
• Matt.. 7:12 - Jesus' golden rule "do unto others" is the converse of Tobit 4:15 - what you hate, do not do to others.
• Matt. 7:16,20 - Jesus' statement "you will know them by their fruits" follows Sirach 27:6 - the fruit discloses the cultivation.
• Matt. 9:36 - the people were "like sheep without a shepherd" is same as Judith 11:19 - sheep without a shepherd.
• Matt. 11:25 - Jesus' description "Lord of heaven and earth" is the same as Tobit 7:18 - Lord of heaven and earth.
• Matt. 12:42 - Jesus refers to the wisdom of Solomon which was recorded and made part of the deuterocanonical books.
• Matt. 16:18 - Jesus' reference to the "power of death" and "gates of Hades" references Wisdom 16:13.
• Matt. 22:25; Mark 12:20; Luke 20:29 - Gospel writers refer to the canonicity of Tobit 3:8 and 7:11 regarding the seven brothers.
• Matt. 24:15 - the "desolating sacrilege" Jesus refers to is also taken from 1 Macc. 1:54 and 2 Macc. 8:17.
• Matt. 24:16 - let those "flee to the mountains" is taken from 1 Macc. 2:28. Matt. 27:43 - if He is God's Son, let God deliver him from His adversaries follows Wisdom 2:18.
• Mark 4:5,16-17 - Jesus' description of seeds falling on rocky ground and having no root follows Sirach 40:15.
• Mark 9:48 - description of hell where their worm does not die and the fire is not quenched references Judith 16:17.

The Deuterocanon quoted (or referred to) in the New Testament

Other answers give lists, but here's a quick one with the cross-references in the Protestant King James Bible (1611).

• Matthew 6:14-15 and Sirach 7:14
• Matthew 27:43 and Wisdom 2:15,16
• Luke 6:31 and Tobit 4:15
• Luke 14:13 and Tobit 4:7
• John 10:22 and 1 Maccabees 4:59
• Romans and Wisdom, clay and the potter
• Romans 11:34 and Wisdom 9:13
• 2 Corinthians 9:7 and Sirach 35:8
• Hebrew 1:3 and Wisdom 7:26
• Hebrews 11:35 and 2 Maccabees 7:7
• Revelation 8:2 and Tobit 12:15 [this one's not in the KJV, but worth noting, seven angels standing before God is not mentioned anywhere else in the Greek Old Testament]

A quotation in the New Testament (or OT) does not make a work part of canon

This is important to note. There are numerous examples from scripture of quotation of religious works that are not canon.

• The usual example is the Book of Enoch, with the passage 1 Enoch 1:9 quoted in Jude 1:14-15
• The Book of Jasher is quoted in Joshua 10:13, 2 Samuel 1:17-19, and 2 Timothy 3:8.
• Book of Shemaiah, and of Iddo the Seer is quoted in 2 Chronicles 9:29, 2 Chronicles 12:15, and 2 Chronicles 13:22
• Book of the Wars of the Lord is quoted in Numbers 21:14

Absence of citation is no argument against canonicity

If absence of New Testament quotations is intended to prove the Deuterocanon as non-canonical, then the same principle would need to be applied the Protocanonical books of Judges, Ruth, Ezra, Nehemiah, Obadiah, Nahum, Esther, the Song of Songs, and Ecclesiastes - all of which are also not quoted in the New Testament, yet both Catholics and Protestants believe those books are part of canon.

Some scriptures quoted in the New Testament have no corresponding Old Testament passage

One interesting thing you may note is that there are works that are explicitly quoted as scripture in the New Testament which don't appear in the Old Testament at all.

Some examples:

• James 4:5 (Scripture says)
• John 7:38 (Scripture has said)
• Matthew 2:23 (spoken by the prophets)
• 1 Corinthians 15:45 (it is written)
• Luke 24:46 (Scriptures, it is written - could be vague allusion to Hosea 6:2)
• Mark 9:12 (is it written - could be vague allusion to Isaiah 53)
• 1 Corinthians 2:9 (it is written - Origen and Jerome identify this as from Apocalypse of Elias)
• Hebrews 11:37 ("sawn in two" - no reference in OT to such an event, ancient tradition relates that Isaiah was thus put to death by order of Manasseh. Justin Martyr mentions this in his Dialogue with Trypho, Chapter 120)

This is interesting, because it gives credence to a current of thought among early Church Fathers that the Jews of their day, who denied Jesus was Christ, had removed scriptures from the Old Testament - and this is what resulted in their reduced canon, smaller than the canon of the Septuagint (which included the Deuterocanon).

This is relevant to your question, as you appear to be focusing on reasons for the inclusion of the deuterocanonical books as canon, but some of the answers you are looking for may be hidden behind how some books got excluded from the canon.

Here Justin Martyr identifies two passages that were in his Old Testament, but were missing from the Jewish scriptures of his day (and from modern Old Testaments).

~160 AD: Justin Martyr in his Dialogue with Trypho, Chapter 72

From the statements, then, which Esdras made in reference to the law of the passover, they have taken away the following: 'And Esdras said to the people, This passover is our Savior and our refuge. And if you have understood, and your heart has taken it in, that we shall humble Him on a standard, and thereafter hope in Him, then this place shall not be forsaken for ever, says the God of hosts. But if you will not believe Him, and will not listen to His declaration, you shall be a laughing-stock to the nations.' And from the sayings of Jeremiah they have cut out the following: 'I [was] like a lamb that is brought to the slaughter: they devised a device against me, saying, Come, let us lay on wood on His bread, and let us blot Him out from the land of the living; and His name shall no more be remembered.' [Jeremiah 11:19] And since this passage from the sayings of Jeremiah is still written in some copies [of the Scriptures] in the synagogues of the Jews (for it is only a short time since they were cut out), and since from these words it is demonstrated that the Jews deliberated about the Christ Himself, to crucify and put Him to death, He Himself is both declared to be led as a sheep to the slaughter, as was predicted by Isaiah, and is here represented as a harmless lamb; but being in a difficulty about them, they give themselves over to blasphemy. And again, from the sayings of the same Jeremiah these have been cut out: 'The Lord God remembered His dead people of Israel who lay in the graves; and He descended to preach to them His own salvation.'

Here, Justin Martyr notes in general that the Jews have taken away many scriptures from the Septuagint ("seventy elders"), resulting in their reduced, smaller canon.

~160 AD: Justin Martyr in his Dialogue with Trypho, Chapter 71

But I am far from putting reliance in your teachers, who refuse to admit that the interpretation made by the seventy elders who were with Ptolemy [king] of the Egyptians is a correct one; and they attempt to frame another. And I wish you to observe, that they have altogether taken away many Scriptures from the translations effected by those seventy elders who were with Ptolemy, and by which this very man who was crucified is proved to have been set forth expressly as God, and man, and as being crucified, and as dying; but since I am aware that this is denied by all of your nation, I do not address myself to these points, but I proceed to carry on my discussions by means of those passages which are still admitted by you. For you assent to those which I have brought before your attention, except that you contradict the statement, 'Behold, the virgin shall conceive,' and say it ought to be read, 'Behold, the young woman shall conceive.' And I promised to prove that the prophecy referred, not, as you were taught, to Hezekiah, but to this Christ of mine: and now I shall go to the proof...

Origen notes here the differences between the Greek copy of the Old Testament in every Church of Christ, and the Jewish copy of the Old Testament from the Jews in his day. He believes the Jewish rulers and elders took away every passage which might bring them into discredit among the people, giving examples like the story of Susanna which is in the Septuagint but not the Jewish copy.

~240 AD: Origen in his Letter to Africanus

In answer to this, I have to tell you what it behooves us to do in the cases not only of the History of Susanna, which is found in every Church of Christ in that Greek copy which the Greeks use, but is not in the Hebrew, or of the two other passages you mention at the end of the book containing the history of Bel and the Dragon, which likewise are not in the Hebrew copy of Daniel; but of thousands of other passages also which I found in many places when with my little strength I was collating the Hebrew copies with ours. [...]

And I make it my endeavor not to be ignorant of their various readings, lest in my controversies with the Jews I should quote to them what is not found in their copies, and that I may make some use of what is found there, even although it should not be in our Scriptures. [...]

Wherefore I think no other supposition is possible, than that they who had the reputation of wisdom, and the rulers and elders, took away from the people every passage which might bring them into discredit among the people. We need not wonder, then, if this history of the evil device of the licentious elders against Susanna is true, but was concealed and removed from the Scriptures by men themselves not very far removed from the counsel of these elders. [...]

What I have said is, I think, sufficient to prove that it would be nothing wonderful if this history were true, and the licentious and cruel attack was actually made on Susanna by those who were at that time elders, and written down by the wisdom of the Spirit, but removed by these rulers of Sodom, as the Spirit would call them. [...]

Here Jerome and Augustine discuss that Jews had omitted or tampered scriptures. Augustine makes clear the difference between Jews before the time of Christ, and Jews after the time of Christ who denied Jesus was the Christ.

~400 AD: Jerome & Augustine in their letters to each other

Jerome, Letter 75: ...In my attempt to translate into Latin, for the benefit of those who speak the same language with myself, the corrected Greek version of the Scriptures, I have labored not to supersede what has been long esteemed, but only to bring prominently forward those things which have been either omitted or tampered with by the Jews, in order that Latin readers might know what is found in the original Hebrew.

Augustine, Letter 82: As to your translation, you have now convinced me of the benefits to be secured by your proposal to translate the Scriptures from the original Hebrew, in order that you may bring to light those things which have been either omitted or perverted by the Jews. But I beg you to be so good as state by what Jews this has been done, whether by those who before the Lord's advent translated the Old Testament--and if so, by what one or more of them--or by the Jews of later times, who may be supposed to have mutilated or corrupted the Greek Mss., in order to prevent themselves from being unable to answer the evidence given by these concerning the Christian faith. I cannot find any reason which should have prompted the earlier Jewish translators to such unfaithfulness. I beg of you, moreover, to send us your translation of the Septuagint, which I did not know that you had published.

And here Isidore says the Jews originally received the Book of Wisdom (one of the Deuterocanon) as part of their canonical scriptures, but after seeing Jesus Christ and recognizing the prophecies in that book, removed it from their canon.

~600 AD: Isidore of Seville

As a certain one of those who know has recorded, the Hebrews received this work (Wisdom) among the Canonical Scriptures. But after they had seized and killed the Christ, remembering the most evident testimonies concerning Christ in that same book, in which it is written: ‘The impious said among themselves, ‘let us seize the just,’ etc., taking counsel lest we might lay upon them such an evident sacrilege, they cut it off from the prophetic volumes, and prohibited its reading to their people. [quoted by Andrew Edward Breen in his A General and Critical Introduction to the Study of Holy Scripture. Original source not identified.]

References for the above, and more details, are available here