According to Acts 10:43, how does salvation come — through Jesus' name or through His work on the cross?

Les explains that Peter was in the home of Cornelius, preaching to Gentiles. While Peter was yet preaching, the Holy Spirit came upon those Gentiles, evidenced with tongues, and Peter was flabbergasted because the expected order had not occurred. Les points out that Peter had not said a word about the work of the cross for salvation by his shed blood, like Paul would have done. Instead, Peter rehearsed that Jesus of Nazareth had been crucified, killed, raised from the dead, and was alive. Then in Acts 10:43, Peter says all the prophets witness that through His Name—not through His work of the cross—is salvation. Les stresses that Peter as yet did not know Paul's Gospel, but Peter did know that the Name of Jesus of Nazareth, the Messiah, the Son of God, was the vehicle for salvation.