Like you said in your question, there is the Magnificat, also known as the Song of Mary.

Luke 1:46-55 ESV
And Mary said,
“My soul magnifies the Lord,
and my spirit rejoices in God my Savior,
for he has looked on the humble estate of his servant.
For behold, from now on all generations will call me blessed;
for he who is mighty has done great things for me,
and holy is his name.
And his mercy is for those who fear him
from generation to generation.
He has shown strength with his arm;
he has scattered the proud in the thoughts of their hearts;
he has brought down the mighty from their thrones
and exalted those of humble estate;
he has filled the hungry with good things,
and the rich he has sent away empty.
He has helped his servant Israel,
in remembrance of his mercy,
as he spoke to our fathers,
to Abraham and to his offspring forever.”

The Book of Revelation written by a prophet of God, a seer who was every bit the equal of any OT prophet, is full of rich imagery equal to any artists' rendering we might go to a gallery to see. The book is a canvas that John painted on and a tablet that a poet set down verse and couplet upon. In the case of the book of Revelation, John provided the paintbrush and the pen and ink for the artist and poet who was God Himself.

Examples:

Revelation 21:2. And I John saw the holy city, new Jerusalem, coming down from God out of heaven, prepared as a bride adorned for her husband.

Revelation 14:6. And I saw another angel fly in the midst of heaven, having the everlasting gospel to preach unto them that dwell on the earth, and to every nation, and kindred, and tongue, and people,

Revelation 12 King James Version (KJV) 12 And there appeared a great wonder in heaven; a woman clothed with the sun, and the moon under her feet, and upon her head a crown of twelve stars: 2 And she being with child cried, travailing in birth, and pained to be delivered. 3 And there appeared another wonder in heaven; and behold a great red dragon, having seven heads and ten horns, and seven crowns upon his heads. 4 And his tail drew the third part of the stars of heaven, and did cast them to the earth: and the dragon stood before the woman which was ready to be delivered, for to devour her child as soon as it was born. 5 And she brought forth a man child, who was to rule all nations with a rod of iron: and her child was caught up unto God, and to his throne.

The Song of Moses;

3And they sang the song of Moses, the bond-servant of God, and the song of the Lamb, saying, "Great and marvelous are Your works, O Lord God, the Almighty; Righteous and true are Your ways, King of the nations! 4"Who will not fear, O Lord, and glorify Your name? For You alone are holy; For ALL THE NATIONS WILL COME AND WORSHIP BEFORE YOU, FOR YOUR RIGHTEOUS ACTS HAVE BEEN REVEALED."…

Mary and Simeon were prophesying. It is fairly common for Biblical prophecy (not the predictive kind) to be expressed in poetic language. However, the OT has whole books consisting entirely of poetry. The NT has only few passages here and there, hence there are no "poetic books" of the NT as there are of the OT.

I can think of one verse that is recited by Apostle Paul. Acts 17:28 states, “In him we live, and move, and have our being as certain also of your own poets have said, For we are also his offspring.” Poetry was given in the Old Testament as an expression of the heart, mostly found in David’s heart, and that heart was also seen in Solomon. I think that we see more poetry in the Old Testament, because it is true that the Old Testament is the New Testament concealed, and the New Testament is the Old Testament revealed. When Christ came he was the revelation and completeness and fulness of everyman’s heart, just as Paul stated. Outside of this quoted poem I know of none else.

It would appear to me that 1 John 4:7-21 is a discourse on love very much in the style of Hebrew poetry. The parallel thoughts in verse 7: "Let us love"; verse 11: "We ought to love"; and verse 21, where we are commanded to love, provide the frame on which are hung numerous pairs and triads of parallel thoughts. Sidlow Baxter in "Explore the Book" notes "to understand this Hebrew parallelism is not only poetically interesting; it is important in the interpretation of Scripture. The corresponding members in each parallel throw light upon each other". With that in mind note how the parallel thoughts in this passage enlighten each other.

God Is Love 1 John 4:7-21

A1 Why Love?

7 Beloved, let us love one another, for love is from God; (Reason to love)

B1 What love proves

and everyone who loves is [a]born of God and knows God.(Positive) 8 The one who does not love does not know God, for God is love.(Negative) (Love sign of knowing God)

C1 Love’s History

9 By this the love of God was manifested [b]in us, that God has sent His [c]only begotten Son into the world so that we might live through Him.(Love shown) 10 In this is love, not that we loved God, but He loved us and sent His Son to be the propitiation for our sins. (Love defined)

A2 Why Love?

11 Beloved, if God so loved us, we also ought to love one another. (Obligation to love)

B2 What love proves

12 No one has seen God at any time; if we love one another, God abides in us, and His love is perfected in us. (Love sign of God’s abiding in us)

D1-D3 Love’s Witness

13 By this we know that we abide in Him and He in us, because He has given us of His Spirit. (Inner witness) 14 We have seen and testify that the Father has sent the Son to be the Savior of the world. (Apostolic witness) 15 Whoever confesses that Jesus is the Son of God, God abides in him, and he in God. (Public witness)

B3 What love proves

16 We have come to know and have believed the love which God has [d]for us. God is love, and the one who abides in love abides in God, and God abides in him. (Love sign of us abiding in God)

C2 Love’s History

17 By this, love is perfected with us, so that we may have confidence in the day of judgement; because as He is, so also are we in this world. (L. perfected confidence) 18 There is no fear in love; but perfect love casts out fear, because fear [e]involves punishment, and the one who fears is not perfected in love. (L. perfected no fear)

B4 What love proves

19 We love, because He first loved us. (Principle) 20 If someone says, “I love God,” and hates his brother, he is a liar; for the one who does not love his brother whom he has seen, cannot love God whom he has not seen. (Loving others sign of loving God) (Example)

A3 Why Love?

21 And this commandment we have from Him, that the one who loves God should love his brother also. (Command to Love)

1 Corinthians chapter 13 is called the Hymn of Love and is thought to be a pre-Pauline hymn that Paul was quoting. Perhaps the most beautiful English rendition of the poem or song is the the King James Bible, but unfortunately it translates ἀγάπην as 'charity', rather that 'love'. The Hymn of Love, from the KJV, amended to speak of 'love':

1 Though I speak with the tongues of men and of angels, and have not love, I am become as sounding brass, or a tinkling cymbal.

2 And though I have the gift of prophecy, and understand all mysteries, and all knowledge; and though I have all faith, so that I could remove mountains, and have not love, I am nothing.

3 And though I bestow all my goods to feed the poor, and though I give my body to be burned, and have not love, it profiteth me nothing.

4 Love suffereth long, and is kind; love envieth not; love vaunteth not itself, is not puffed up,

5 Doth not behave itself unseemly, seeketh not her own, is not easily provoked, thinketh no evil;

6 Rejoiceth not in iniquity, but rejoiceth in the truth;

7 Beareth all things, believeth all things, hopeth all things, endureth all things.

8 Love never faileth: but whether there be prophecies, they shall fail; whether there be tongues, they shall cease; whether there be knowledge, it shall vanish away.

9 For we know in part, and we prophesy in part.

10 But when that which is perfect is come, then that which is in part shall be done away.

11 When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.

12 For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known.

13 And now abideth faith, hope, love, these three; but the greatest of these is love.

1 Peter 3:18-22, which Burton L. Mack, in Who Wrote the New Testament, page 91, calls a ‘Christ hymn’, and says is a genre that became popular among early Christians.

18 For Christ also hath once suffered for sins, the just for the unjust, that he might bring us to God, being put to death in the flesh, but quickened by the Spirit:

19 By which also he went and preached unto the spirits in prison;

20 Which sometime were disobedient, when once the longsuffering of God waited in the days of Noah, while the ark was a preparing, wherein few, that is, eight souls were saved by water.

21 The like figure whereunto even baptism doth also now save us (not the putting away of the filth of the flesh, but the answer of a good conscience toward God,) by the resurrection of Jesus Christ:

22 Who is gone into heaven, and is on the right hand of God; angels and authorities and powers being made subject unto him.

A form of ancient poetry, quite different to modern ideas of poetry was the chiasm (from the Greek letter chi, χ), or chiastic structure. In these structures, an opening set of events is mirrored by a second set. They were used extensively in the Old Testament and, to a lesser extent, in the New Testament, but perhaps no more so than in Mark's Gospel. Here is an example paraphrased from Mark 3:20-4:1 (with three verses of prose interposed at 3:27-29):

a The multitude is so thick that they could not so much as eat bread (3:20)

.b Jesus' friends say he is beside himself (3:21)

..c Scribes say Jesus is possessed and by the devil casts out devils (3:22)

...d How can Satan cast out Satan? (3:23)

....e If a kingdom be divided the kingdom can not stand (3:24)

....e' If a house be divided the house can not stand (3:25)

...d' If Satan is divided he can not win (3:26)

..c' Because they said Jesus has an unclean spirit (3:30)

.b' Jesus' family comes to him. Jesus says his followers are his family (3:31-35)

a' The multitude is so thick that Jesus enters a boat (4:1)

Luke's Magnificat is a poem modelled on Hannah's prayer from the Old Testament. It is the first of four canticles in the opening of Luke’s Gospel and is a beautiful hymn spoken by Mary, at Luke 1:46-55:-

46 And Mary said, My soul doth magnify the Lord,

47 And my spirit hath rejoiced in God my Saviour.

48 For he hath regarded the low estate of his handmaiden: for, behold, from henceforth all generations shall call me blessed.

49 For he that is mighty hath done to me great things; and holy is his name.

50 And his mercy is on them that fear him for all generations.

51 He hath shewed strength with his arm; he hath scattered the proud in the imagination of their hearts.

52 He hath put down the mighty from their seats, and exalted them of low degree.

53 He hath filled the hungry with good things; and the rich he hath sent empty away.

54 He hath holpen his servant Israel, in remembrance of his mercy;

55 As he spake to our fathers, to Abraham, and to his seed for ever.