St. Peter also teaches that by the sacrifice of Christ we are healed from our sins.
1 Peter 2:18-25 (DRB) Servants, be subject to your masters with all fear, not only to the good and gentle, but also to the froward. 19 For this is thankworthy, if for conscience towards God, a man endure sorrows, suffering wrongfully. 20 For what glory is it, if committing sin, and being buffeted for it, you endure? But if doing well you suffer patiently; this is thankworthy before God. 21 For unto this are you called: because Christ also suffered for us, leaving you an example that you should follow his steps. 22 Who did no sin, neither was guile found in his mouth. 23 Who, when he was reviled, did not revile: when he suffered, he threatened not: but delivered himself to him that judged him unjustly. 24 Who his own self bore our sins in his body upon the tree: that we, being dead to sins, should live to justice: by whose stripes you were healed. 25 For you were as sheep going astray; but you are now converted to the shepherd and bishop of your souls.
The bolded portions are a reference to Isaiah 53, which speaks of God's suffering Servant, whom St. Peter authoritatively interprets to be Christ.
Isaiah 53 (DRB) Who hath believed our report? and to whom is the arm of the Lord revealed? 2 And he shall grow up as a tender plant before him, and as a root out of a thirsty ground: there is no beauty in him, nor comeliness: and we have seen him, and there was no sightliness, that we should be desirous of him: 3 Despised, and the most abject of men, a man of sorrows, and acquainted with infirmity: and his look was as it were hidden and despised, whereupon we esteemed him not. 4 Surely he hath borne our infirmities and carried our sorrows: and we have thought him as it were a leper, and as one struck by God and afflicted. 5 But he was wounded for our iniquities, he was bruised for our sins: the chastisement of our peace was upon him, and by his bruises we are healed. 6 All we like sheep have gone astray, every one hath turned aside into his own way: and the Lord hath laid on him the iniquity of us all. 7 He was offered because it was his own will, and he opened not his mouth: he shall be led as a sheep to the slaughter, and shall be dumb as a lamb before his shearer, and he shall not open his mouth. 8 He was taken away from distress, and from judgment: who shall declare his generation? because he is cut off out of the land of the living: for the wickedness of my people have I struck him. 9 And he shall give the ungodly for his burial, and the rich for his death: because he hath done no iniquity, neither was there deceit in his mouth. 10 And the Lord was pleased to bruise him in infirmity: if he shall lay down his life for sin, he shall see a long-lived seed, and the will of the Lord shall be prosperous in his hand. 11 Because his soul hath laboured, he shall see and be filled: by his knowledge shall this my just servant justify many, and he shall bear their iniquities. 12 Therefore will I distribute to him very many, and he shall divide the spoils of the strong, because he hath delivered his soul unto death, and was reputed with the wicked: and he hath borne the sins of many, and hath prayed for the transgressors.
Jesus Himself taught that He came to "give His life a ransom for many," (DRB uses the archaic "redemption" for "ransom"), which appears to be another reference to Isaiah 53, where it says, "he shall lay down his life for sin," and, "he hath borne the sins of many," and is in any case the same teaching.
Matthew 20:25-28 (DRB) But Jesus called them to him, and said: You know that the princes of the Gentiles lord it over them; and they that are the greater, exercise power upon them. 26 It shall not be so among you: but whosoever will be the greater among you, let him be your minister: 27 And he that will be first among you, shall be your servant. 28 Even as the Son of man is not come to be ministered unto, but to minister, and to give his life a redemption for many.
Matthew 26:28 (DRB) ... my blood of the new testament, which shall be shed for many unto remission of sins.
Likewise St. John in 1 John:
1 John 1:5-10 (DRB) And this is the declaration which we have heard from him, and declare unto you: That God is light, and in him there is no darkness. 6 If we say that we have fellowship with him, and walk in darkness, we lie, and do not the truth. 7 But if we walk in the light, as he also is in the light, we have fellowship one with another, and the blood of Jesus Christ his Son cleanseth us from all sin. 8 If we say that we have no sin, we deceive ourselves, and the truth is not in us. 9 If we confess our sins, he is faithful and just, to forgive us our sins, and to cleanse us from all iniquity. 10 If we say that we have not sinned, we make him a liar, and his word is not in us.
This is the same teaching of St. Paul in Romans:
Romans 5:7-11 (DRB) For scarce for a just man will one die; yet perhaps for a good man some one would dare to die. 8 But God commendeth his charity towards us; because when as yet we were sinners, according to the time, 9 Christ died for us; much more therefore, being now justified by his blood, shall we be saved from wrath through him. 10 For if, when we were enemies, we were reconciled to God by the death of his Son; much more, being reconciled, shall we be saved by his life. 11 And not only so; but also we glory in God, through our Lord Jesus Christ, by whom we have now received reconciliation.
And possibly (at least, arguably) not St. Paul in Hebrews:
Hebrews 9:11-14 (DRB) But Christ, being come an high priest of the good things to come, by a greater and more perfect tabernacle not made with hand, that is, not of this creation: 12 Neither by the blood of goats, or of calves, but by his own blood, entered once into the holies, having obtained eternal redemption. 13 For if the blood of goats and of oxen, and the ashes of an heifer being sprinkled, sanctify such as are defiled, to the cleansing of the flesh: 14 How much more shall the blood of Christ, who by the Holy Ghost offered himself unspotted unto God, cleanse our conscience from dead works, to serve the living God?
It is on account of Jesus' sacrifice alone that we have peace with God, according to the entire New Testament.
Of course these quotes only touch the surface.