According to Paul himself, baptism is the moment when we put on Christ, and are buried with him in his death and raised to life with him in his life:
It's the circumcision "made without hands" of the New Covenant:
Colossians 2:11-13 (DRB) In whom also you are circumcised with circumcision not made by hand, in despoiling of the body of the flesh, but in the circumcision of Christ: 12 Buried with him in baptism, in [which] also you are risen again by the faith of the operation of God, who hath raised him up from the dead. 13 And you, when you were dead in your sins, and the uncircumcision of your flesh; he hath quickened together with him, forgiving you all offences:
Romans 6:1-4 (DRB) What shall we say, then? shall we continue in sin, that grace may abound? 2 God forbid. For we that are dead to sin, how shall we live any longer therein? 3 Know you not that all we, who are baptized in Christ Jesus, are baptized in his death? 4 For we are buried together with him by baptism into death; that as Christ is risen from the dead by the glory of the Father, so we also may walk in newness of life.
1 Corinthians 12:12-13 (DRB) For as the body is one, and hath many members; and all the members of the body, whereas they are many, yet are one body, so also is Christ. 13 For in one Spirit were we all baptized into one body, whether Jews or Gentiles, whether bond or free; and in one Spirit we have all been made to drink.
For Paul, and as you shall see the other writers of Scripture, faith in Christ begins with baptism.
Galatians 3:26-27 (DRB) For you are all the children of God by faith, in Christ Jesus. 27 For as many of you as have been baptized in Christ, have put on Christ.
Likewise Acts interprets 'faith in Jesus' primarily as going to get baptized upon feeling convicted about your sins:
Acts 2:37-41 (DRB) Now when they had heard these things, they had compunction in their heart, and said to Peter, and to the rest of the apostles: What shall we do, men and brethren? 38 But Peter said to them: [Repent], and be baptized every one of you in the name of Jesus Christ, for the remission of your sins: and you shall receive the gift of the Holy Ghost. 39 For the promise is to you, and to your children, and to all that are far off, whomsoever the Lord our God shall call. 40 And with very many other words did he testify and exhort them, saying: Save yourselves from this perverse generation. 41 They therefore that received his word, were baptized; and there were added in that day about three thousand souls.
Acts 8:12-13 (DRB) But when they had believed Philip preaching of the kingdom of God, in the name of Jesus Christ, they were baptized, both men and women. 13 Then Simon himself believed also; and being baptized, he adhered to Philip. And being astonished, wondered to see the signs and exceeding great miracles which were done.
Notice that this was so fundamental to their preaching and the faith of the readers that baptism is assumed—you read about it nowhere in the recorded text except for what it must have been expressed by: belief in Jesus.
Acts 16:27-33 (DRB) And the keeper of the prison, awaking out of his sleep, and seeing the doors of the prison open, drawing his sword, would have killed himself, supposing that the prisoners had been fled. 28 But Paul cried with a loud voice, saying: Do thyself no harm, for we all are here. 29 Then calling for a light, he went in, and trembling, fell down at the feet of Paul and Silas. 30 And bringing them out, he said: Masters, what must I do, that I may be saved? 31 But they said: Believe in the Lord Jesus, and thou shalt be saved, and thy house. 32 And they preached the word of the Lord to him and to all that were in his house. 33 And he, taking them the same hour of the night, washed their stripes, and himself was baptized, and all his house immediately.
This isn't something you do when you are a 'mature Christian' and want to associate yourself with Christ. This is the gospel "preached to every creature."
Again, the Ethiopian eunuch has Jesus preached to him by Philip, and the first thing he does after having heard the gospel is point to the nearest body of water in which to be baptized; clearly, baptism was a fundamental part of the gospel according to Philip:
Acts 8:26-39 (DRB) Now an angel of the Lord spoke to Philip, saying: Arise, go towards the south, to the way that goeth down from Jerusalem into Gaza: this is desert. 27 And rising up, he went. And behold a man of Ethiopia, an eunuch, of great authority under Candace the queen of the Ethiopians, who had charge over all her treasures, had come to Jerusalem to adore. 28 And he was returning, sitting in his chariot, and reading Isaias the prophet. 29 And the Spirit said to Philip: Go near, and join thyself to this chariot. 30 And Philip running thither, heard him reading the prophet Isaias. And he said: Thinkest thou that thou understandest what thou readest? 31 Who said: And how can I, unless some man shew me? And he desired Philip that he would come up and sit with him. 32 And the place of the scripture which he was reading was this: He was led as a sheep to the slaughter; and like a lamb without voice before his shearer, so openeth he not his mouth. 33 In humility his judgment was taken away. His generation who shall declare, for his life shall be taken from the earth? 34 And the eunuch answering Philip, said: I beseech thee, of whom doth the prophet speak this? of himself, or of some other man? 35 Then Philip, opening his mouth, and beginning at this scripture, preached unto him Jesus. 36 And as they went on their way, they came to a certain water; and the eunuch said: See, here is water: what doth hinder me from being baptized? 37 And Philip said: If thou believest with all thy heart, thou mayest. And he answering, said: I believe that Jesus Christ is the Son of God. 38 And he commanded the chariot to stand still; and they went down into the water, both Philip and the eunuch: and he baptized him. 39 And when they were come up out of the water, the Spirit of the Lord took away Philip; and the eunuch saw him no more. And he went on his way rejoicing.
According to the Evangelists, too, baptism is the primary way you become a believer in the gospel, and part of the body of Christ—his disciple:
Matthew 28:19 (DRB) Going therefore, teach ye all nations; baptizing them in the name of the Father, and of the Son, and of the Holy Ghost.
Mark 16:15 (DRB) And he said to them: Go ye into the whole world, and preach the gospel to every creature.
Baptism is the new circumcision. It is how one associates themselves to Christ by faith, not circumcising flesh, but the heart.
So, when we get to Paul, the first thing after he sees Jesus, and thus converts, he is immediately baptized by express wish of Jesus.
Act 9:1-18 (DRB) And Saul, as yet breathing out threatenings and slaughter against the disciples of the Lord, went to the high priest, 2 And asked of him letters to Damascus, to the synagogues: that if he found any men and women of this way, he might bring them bound to Jerusalem.
3 And as he went on his journey, it came to pass that he drew nigh to Damascus; and suddenly a light from heaven shined round about him. 4 And falling on the ground, he heard a voice saying to him: Saul, Saul, why persecutest thou me? 5 Who said: Who art thou, Lord? And he: I am Jesus whom thou persecutest. It is hard for thee to kick against the goad. 6 And he trembling and astonished, said: Lord, what wilt thou have me to do? 7 And the Lord said to him: Arise, and go into the city, and there it shall be told thee what thou must do. Now the men who went in company with him, stood amazed, hearing indeed a voice, but seeing no man. 8 And Saul arose from the ground; and when his eyes were opened, he saw nothing. But they leading him by the hands, brought him to Damascus. 9 And he was there three days, without sight, and he did neither eat nor drink.
10 Now there was a certain disciple at Damascus, named Ananias. And the Lord said to him in a vision: Ananias. And he said: Behold I am here, Lord. 11 And the Lord said to him: Arise, and go into the street that is called Strait, and seek in the house of Judas, one named Saul of Tarsus. For behold he prayeth. 12 (And he saw a man named Ananias coming in, and putting his hands upon him, that he might receive his sight.) 13 But Ananias answered: Lord, I have heard by many of this man, how much evil he hath done to thy saints in Jerusalem. 14 And here he hath authority from the chief priests to bind all that invoke thy name. 15 And the Lord said to him: Go thy way; for this man is to me a vessel of election, to carry my name before the Gentiles, and kings, and the children of Israel. 16 For I will shew him how great things he must suffer for my name's sake.
17 And Ananias went his way, and entered into the house. And laying his hands upon him, he said: Brother Saul, the Lord Jesus hath sent me, he that appeared to thee in the way as thou camest; that thou mayest receive thy sight, and be filled with the Holy Ghost. 18 And immediately there fell from his eyes as it were scales, and he received his sight; and rising up, he was baptized.
I want to direct attention to the timing here. Paul has just seen Jesus, and is repenting, not eating or drinking, confused, scared, most probably. The first time he sees the door open, so to speak, is a man who comes to heal and then baptized him—immediately.
In other words, according to Paul and other Scriptures, baptism is the new circumcision. It's how you become part of the New Covenant, buried with Christ into His death and thus a partaker in His life. Paul was justified when he as baptized into Christ's death. But regeneration does not necessitate freedom to sin is taken away, and on that account, late in his life, Paul says that if he had done something wrong, he would not be justified: "But to me it is a very small thing to be judged by you, or by man's day; but neither do I judge my own self. 4 For I am not conscious to myself of any thing, yet am I not hereby justified; but he that judgeth me, is the Lord" (1 Corinthians 4:3-4; see also 10:1-12).
Likewise, St. Peter teaches rather directly, in teaching how baptism is prefigured in a way by the purging waters of the Flood.
1 Peter 3:18-21 (DRB) Because Christ also died once for our sins, the just for the unjust: that he might offer us to God, being put to death indeed in the flesh, but enlivened in the spirit, 19 In which also coming he preached to those spirits that were in prison: 20 Which had been some time incredulous, when they waited for the patience of God in the days of Noe, when the ark was a building: wherein a few, that is, eight souls, were saved by water. 21 Whereunto baptism being of the like form, now saveth you also: not the putting away of the filth of the flesh, but the [appeal] of a good conscience toward God by the resurrection of Jesus Christ.
Again, Jesus Himself:
Mark 16:15-16 (DRB) And he said to them: Go ye into the whole world, and preach the gospel to every creature. 16 He that believeth and is baptized, shall be saved: but he that believeth not shall be condemned.
"Believes not and is not baptized" is redundant, according to Jesus; and so believing and being baptized are inseparable.
I think the point is clear. One is initially saved or justified in baptism. Whether he wallows in the mud again, or whatever else, comes after.
This writing from an extremely early Christian is representative of the ancient and universal view of what it meant to be baptized in the early Church also:
We also pray and fast with them [neophytes/new converts]. Then we bring them to a place where there is water, and they are regenerated in the same manner in which we ourselves were regenerated. They then receive the washing with water in the name of God (the Father and Lord of the universe) and of our Savior Jesus Christ, and of the Holy Spirit. For Christ said, ‘Unless you are born again, you shall not enter into the kingdom of heaven’ (First Apology, Chapter 61, A.D. 155).
And,
Wash therefore, and be now clean, and put away iniquity from your souls, as God bids you be washed in this laver, and be circumcised with the true circumcision (Dialogue with Trypho, Chapter 18)