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“The washing away of sins”

"So there was no need for a sacrifice?"

Of course there was. The death of Christ is the cause of the sacrament of baptism. Baptism works by the saving work of Christ.

1225 In his Passover Christ opened to all men the fountain of Baptism. He had already spoken of his Passion, which he was about to suffer in Jerusalem, as a "Baptism" with which he had to be baptized.²² The blood and water that flowed from the pierced side of the crucified Jesus are types of Baptism and the Eucharist, the sacraments of new life.²³ From then on, it is possible "to be born of water and the Spirit"²⁴ in order to enter the Kingdom of God.

See where you are baptized, see where Baptism comes from, if not from the cross of Christ, from his death. There is the whole mystery: he died for you. In him you are redeemed, in him you are saved.²⁵

22 Mk 10:38; cf. Lk 12:50.
23 Cf. Jn 19:34; 1 Jn 5:6-8.
24 Cf. Jn 3:5.
25 St. Ambrose, De sacr. 2, 2, 6: PL 16, 444; cf. Jn 3:5.

Source: CCC