Stoke broke the deadlock with the first meaningful effort on goal in the 24th minute. Matthew Etherington curled in a corner for centre-back Upson to head into the net from eight yards out to claim his first goal for the club.
The home side went on to double their advantage six minutes before the break. Ryan Shotton's long throw was met by Crouch, who sent his header past debutant goalkeeper Gerhard Tremmel.
Stoke continued to dominate after the break and almost added a third in the 61st minute. Shotton's long throw again caused problems for the Swansea defence but Jonathan Walters was unable to get his header on target.
It took Swansea until the 76th minute to register their first shot on target when Luke Moore fired straight at Asmir Begovic.
The victory has seen Stoke jump above the Swans in the Premier League table.