The Catholic Church affirms unequivocally that Jesus Christ is the only Savior. Everyone who is saved (that is, who goes to Heaven) does so through Jesus Christ, regardless of whether he is aware (here on earth) that Jesus is his Savior.

The uniqueness and universality of Jesus as Savior stems from the fact that he—and only he—is the Incarnate Son of God, the “one mediator between God and men” (1 Tim. 2:5).

Dominus Iesus

The most important recent document regarding this topic is a document from the Congregation for the Doctrine of the Faith called Dominus Iesus, issued in 2000. The document states:

In fact, the truth of Jesus Christ, Son of God, Lord and only Saviour, who through the event of his incarnation, death and resurrection has brought the history of salvation to fulfilment, and which has in him its fullness and centre, must be firmly believed (firmiter credenda est) as a constant element of the Church’s faith (No. 13, emphasis in original).

Also, in number 14:

It must therefore be firmly believed (firmiter credenda) as a truth of Catholic faith that the universal salvific will of the One and Triune God is offered and accomplished once for all in the mystery of the incarnation, death, and resurrection of the Son of God.

The phrase firmiter credenda is a technical term that means that the uniqueness of Christ’s salvation is dogma, which must be believed with divine and Catholic faith. (See the doctrinal commentary on the motu proprio Ad tuendam fidem for an explanation of these terms, especially the section that speaks of doctrines de fide credenda, doctrines “that stem from the faith to be believed.”)

The Catechism of the Catholic Church, quoting extensively from Lumen gentium 14, teaches the same doctrine in different terms in number 846:

How are we to understand this affirmation [i.e., that there is no salvation outside the Church], often repeated by the Church Fathers? Re-formulated positively, it means that all salvation comes from Christ the Head through the Church which is his Body:

Basing itself on Scripture and Tradition, the Council teaches that the Church, a pilgrim now on earth, is necessary for salvation: the one Christ is the mediator and the way of salvation; he is present to us in his body which is the Church. He himself explicitly asserted the necessity of faith and Baptism, and thereby affirmed at the same time the necessity of the Church which men enter through Baptism as through a door. Hence they could not be saved who, knowing that the Catholic Church was founded as necessary by God through Christ, would refuse either to enter it or to remain in it [Lumen Gentium 14].

(The numbers that follow, however, can be interpreted somewhat ambiguously, which is doubtless one reason why the CDF felt the need to issue a clarification. See below.)

What this doctrine implies for those who do not know Christ

As KorvinStarmast’s answer points out, the Church does not teach that there is no hope of salvation for those who are not members of the Church, even for those who (through no fault of their own) do not know Jesus Christ. As the Catechism says,

847. This affirmation [i.e., that there is no salvation outside the Church] is not aimed at those who, through no fault of their own, do not know Christ and his Church:

Those who, through no fault of their own, do not know the Gospel of Christ or his Church, but who nevertheless seek God with a sincere heart, and, moved by grace, try in their actions to do his will as they know it through the dictates of their conscience - those too may achieve eternal salvation [Lumen gentium 16].

848. “Although in ways known to himself God can lead those who, through no fault of their own, are ignorant of the Gospel, to that faith without which it is impossible to please him, the Church still has the obligation and also the sacred right to evangelize all men” [Decree Ad gentes on the Mission Activity of the Church, 7].

(These are the numbers I mentioned that admit of some ambiguity. There is nothing wrong with what they say, but the opening phrase of number 847, “This affirmation is not aimed...,” could be taken to refer to the entire number 846 or else only to the doctrine that “outside the Church there is no salvation.” With the clarifications of Dominus Iesus, it is clear that the second interpretation is the correct one. There are no exceptions to the uniqueness of Christ as Savior.)

Hence, although non-Catholics, and even non-Christians, have the possibility of being saved, their salvation is always brought about by Jesus Christ, whether they realize it or not.

I should point out that the Church teaches that all men have the possibility of being saved; no one (not even Catholics) has a guarantee of being saved. Moreover, in the Catholic Church’s view, in the Catholic Church alone are all the means of salvation to be found:

For it is only through Christ's Catholic Church, which is “the all-embracing means of salvation,” that they can benefit fully from the means of salvation (Vatican II’s decree on ecumenism Unitatis redintegratio, 3).

Hence, there is still (in the Church’s view) an objective obligation for all men to belong to the Church. If, however, though no fault on his own, a person is not in a position to be a member of the Church, God will not, of course, hold it against him.

In summary

• Jesus Christ is the unique and universal Savior. Everyone who is saved, is saved through Christ (whether the person realizes it or not).
• There is no other path to salvation other than Christ, who is “the way, and the truth, and the life” (Jn. 14:6) and the “one mediator between God and men” (1 Tim. 2:5).
• Those who, through no fault of their own, do not know Christ, or do not belong fully to the Church that he established, nevertheless have the possibility of salvation. That salvation, however, is always through Christ.

But can you say that that is absolutely and exclusive or do you have to leave an opening because we don't always know the ways of God?

Current teaching leans more toward the latter than the former ... but the simplest (and arguably the best) way is through Christ. (CCC 1987 - 2029 is a discourse to that effect, in detail, as is the treatment in CCC 1023 - 1029). A reaffirmation of salvation being through Jesus is found in Dominus Iesus. (@AthanasiusofAlex goes into a detailed analysis of that source in his answer). The takeaway from that is that if you know of Christ and don't accept the invitation for salvation through him, then having a relationship and a friendship with God is at serious risk.

Discussion

In current teaching the general case is that through Christ is the path to salvation. The Roman Catholic Church allows that there may be salvation subject to the Grace, Mercy (and discretion) of God for those in a relationship of friendship with God who through no fault of their own don't know Christ. But even then, it was Christ who opened the path back to Heaven, so salvation has been through him in the meta sense.

CCC 1026 By his death and Resurrection, Jesus Christ has "opened" heaven to us. The life of the blessed consists in the full and perfect possession of the fruits of the redemption accomplished by Christ.

Grace per CCC 1996:

Grace is favor, the free and undeserved help that God gives us to respond to his call to become children of God, adoptive sons, partakers of the divine nature and of eternal life.

God can give his grace to whomever He so chooses. How someone responds to that gift of grace will inform their relationship with God.

1. From the Catechism of the Catholic Church, article 1023, you will find this statement in a discussion about "Particular Judgment":

   1023 Those who die in God's grace and friendship and are perfectly purified live for ever with Christ. They are like God for ever, for they "see him as he is," face to face:

   1025 To live in heaven is "to be with Christ." the elect live "in Christ," but they retain, or rather find, their true identity, their own name.

2. From the Vatican II council, Lumen Gentium, 16 we find:

   16. Finally, those who have not yet received the Gospel are related in various ways to the people of God. In the first place we must recall the people to whom the testament and the promises were given and from whom Christ was born according to the flesh. On account of their fathers this people remains most dear to God, for God does not repent > of the gifts He makes nor of the calls He issues. But the plan of > salvation also includes those who acknowledge the Creator. In the first place amongst these there are the Muslims, who, professing to hold the faith of Abraham, along with us adore the one and merciful God, who on the last day will judge mankind. Nor is God far distant from those who in shadows and images seek the unknown God, for it is He who gives to all men life and breath and all things, and as Saviour wills that all men be saved. Those also can attain to salvation who through no fault of their own do not know the Gospel of Christ or His Church, yet sincerely seek God and moved by grace strive by their deeds to do His will as it is known to them through the dictates of conscience. Nor does Divine Providence deny the helps necessary for salvation to those who, without blame on their part, have not yet arrived at an explicit knowledge of God and with His grace strive to live a good life. Whatever good or truth is found amongst them is looked upon by the Church as a preparation for the Gospel. She knows that it is given by Him who enlightens all men so that they may finally have life. But often men, deceived by the Evil One, have become vain in their reasonings and have exchanged the truth of God for a lie, serving the creature rather than the Creator. Or some there are who, living and dying in this world without God, are exposed to final despair. Wherefore to promote the glory of God and procure the salvation of all of these, and mindful of the command of the Lord, "Preach the Gospel to every creature", the Church fosters the missions with care and attention.

To put that all together:

• There is a general principle in Christianity (which includes the RCC) that God's grace is required to get into Heaven.

• The usual path to God's grace is via Christ. (The general case for Christian denominations, not just the RCC as explained in CCC 1987-2029 cited abpve).

• The Catechism states that if you die in God's grace and friendship, you'll meet Christ in Heaven. (This opens the door for Lumen Gentium's point, or comes from it, as LG was written before the current catechism was published).

• Lumen Gentium allows that some other people are expected to be in friendship with God and even in His grace. Even so, it was via Christ's agency that the path to heaven is even open. That makes all salvation through Christ, in the larger sense.

• The Church has a duty to get as many people as can be to receive Christ and take that general path toward God's grace.

Without turning this into a nine page long post, the Church teaches (as do most Christians I have engaged on this topic) that forming a relationship with Christ is the direct path to being open to receiving God's grace. I've had that point explained to me by a variety of clergy and lay people in the Roman Catholic faith, as well a Methodists and Episcopals ... just off the top of my head. There have been others. This core point underwrites the longer discussion in the Catechism (CCC 1026) about Christ opening the way to Heaven.