The Dutch midfielder scored two goals to condemn the visitors to their third straight league defeat, while Ben Davies netted his first ever strike for the Swans to stay in eighthSwansea are unbeaten in six Premier League games after a fine 3-1 win over Stoke at the Liberty Stadium, with Jonathan De Guzman netting twice to keep the Swans in eighth place in the table.
Ben Davies opened the scoring with an excellent goal early in the second half before the Dutch midfielder Guzman added two quality strikes to seal the result.
Former England striker Michael Owen scored late on with his first ever goal for the Potters, but it was too little, too late for Tony Pulis' side, who stay in 10th position.
Laudrup made eight changes from the loss to Arsenal midweek, with Michel Vorm, De Guzman and Wayne Routledge the only survivors.
Asmir Begovic, Glenn Whelan and Charlie Adam all returned for Stoke with Dean Whitehead filling in at right-back while Kenwyne Jones, a Swansea target, was among the substitutes.
The Swans 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.
Stoke began to fustrate the hosts as the match went on and were struggling to get men ahead of the ball with Peter Crouch cutting an isolated figure in attack.
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 could not sort out 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 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 close-range header.
The second half sparked into life instantly when Davies scored with a superbly-taken first goal. The left-back shrugged off Jonathan Walters before skipping past Robert Huth and slotting a low right-footed finish calmly 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 curl a sweet 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 the midfielder provided a finish to match.
Owen grabbed a late consolation for Stoke with his first goal for the club as a late substitute, getting a slight touch on Cameron Jerome's cross to take it into the net.