Welsch was drafted by the NBA's Philadelphia 76ers with the 16th selection of the 2002 NBA draft, but he was immediately traded to the Golden State Warriors for a future first-round draft pick and a future first- or second-round draft pick. He spent one season with the Warriors, averaging 1.6 points per game, 0.8 rebounds per game, and 0.7 assists per game in 37 games in the 2002–03 season before he was traded to the Dallas Mavericks, along with Antawn Jamison, Chris Mills and Danny Fortson, in exchange for Evan Eschmeyer, Nick Van Exel, Avery Johnson, Popeye Jones and Antoine Rigaudeau on 18 August 2003. Without ever playing a game for them, Welsch, along with Mills, Raef LaFrentz and a lottery-protected 2004 NBA Draft pick, was traded by Dallas to the Boston Celtics on 20 October 2003 in exchange for Antoine Walker and Tony Delk. Welsch started 68 games for the Celtics in the 2003–04 season, averaging 9.2 points per game.