Actually, it is the baptism done BY JESUS - the baptism of the Holy Spirit - that saves us and gives us new life. It is by the baptism of the Spirit that we are placed “In Christ”, and we are only saved in Christ:
“12 For just as the body is one and has many members, and all the members of the body, though many, are one body, so it is with Christ. 13 For in one Spirit we were all baptized into one body—Jews or Greeks, slaves or free—and all were made to drink of one Spirit.” - 1 Corinth 12
Exactly. It’s the Holy Spirit that baptizes, not the water. The thief on the cross WAS baptized.