Ben Davies scored his first ever goal for the club before the Dutchman added two more to condemn the visitors to their third straight league defeat
Ben Davies opened the scoring with an excellent goal early in the second half before Jonathan de Guzman added two quality goals to seal the result.
Michael Owen added a late consolation with his first ever strike for the Potters.
Swansea started the brighter of the two teams, with Pablo Hernandez dragging a shot wide of Begovic's goal in the opening minutes, before Michu miscued a volley from just outside the area after a good lay-off from Itay Schecter.
The hosts' first real chance of the half came after 26 minutes when Hernandez found Michu with a long ball into the box but the Spaniard couldn't rerarrange his feet and sent his volley off target.
That was quickly followed by a chance for the visitors. Matthew Etherington should have done better as he failed to hit the target from a tight angle inside the box following a neat exchange with Adam.
Etherington then nearly handed the Swans a lead in the 36th minute when he over-played deep inside his own half, allowing Angel Rangel to dispossess him before crossing for Michu, who should have scored with his header inside the six-yard box.
The second half sparked instantly into life when Davies gave Swansea the lead with a superbly taken first goal for the club. The left back shrugged off Jonathan Walters before skipping past Robert Huth and slotting a low right-footed finish past Begovic.
Less than 10 minutes later the hosts doubled their lead through a fantastic free kick from De Guzman. Michu was fouled just outside the box, allowing the Dutchman to step up and curl a sweetly struck right-footed effort into the top corner.
The Dutchman got himself on the scoresheet again with 10 minutes to play. He made a lung-busting run to support Danny Graham, whose lay-off was sublimely controlled and De Guzman provided a finish to match.
Owen grabbed a late consolation for Stoke with his first goal for the club as a late substitute, the striker got a slight touch on Cameron Jerome's cross to take it into the net.