Second-half strikes from Waye Routledge, Nathan Dyer and Wilfried Bony secured all three points for the home side, as the Swans avenged their 1-0 defeat to Cardiff earlier this season in stunning fashion.
Garry Monk was installed as caretaker coach in the wake of Laudrup's dismissal and this win did the former Swansea captain's chances of becoming the Dane's long-term successor no harm at all.
Cardiff was a 1-0 victor on its own patch in November in the Premier League but boss Ole Gunnar Solskjaer was left to rue a pair of first-half misses from Kenwyne Jones, while Craig Bellamy rattled the crossbar at 1-0.
Monk handed Marvin Emnes a second Swansea debut up front following the Middlesbrough forward's surprise loan signing on deadline day.
Meanwhile, Cardiff's headline-making January acquisition Wilfried Zaha earned a full debut on the back of an impressive cameo in last weekend's win over Norwich City.
Emnes, operating behind central striker Bony, dinked the ball over Ben Turner and drove at the Cardiff defense in the opening minute, playing in Routledge, whose shot was saved by David Marshall.
The Cardiff goalkeeper then judged a challenge on Bony to perfection as the fired-up home faithful howled in vain for a penalty.
Bellamy scored the only goal the last time these sides met at the Liberty Stadium and the veteran forward predictably started to make his presence felt.
Jones soon stole half a yard on Chico Flores to head his strike partner's right-wing cross wide before, in the 24th minute, the same player stabbed off target after Bellamy slickly exchanged passes with Peter Whittingham.
But, having been on the back foot, Swansea ended the half strongly – Dyer flashing a ball across the goalmouth and Jonathan de Guzman glancing wide – though the team was undermined by a lack of cutting edge.
Monk sent on Pablo Hernandez at the break as a thigh problem curtailed Emnes' evening. As first managerial substitutions go, it proved particularly inspired.
In the second minute after the restart, Hernandez measured a through ball to catch Fabio da Silva napping and Routledge produced a clipped finish to open his Premier League account for the season.
Cardiff almost had a 51st-minute response through Bellamy's blurring energy - the former Wales international cutting inside past Flores to thump a 20-yard strike against the crossbar.
Dyer shot wide from a similar distance as the Cardiff defense backed off dangerously five minutes later, while Turner stood firm to divert a Bony's volley on the turn wide as the game entered its final quarter.
Then, Kim Bo-Kyung hooked over when Swansea failed to clear a set piece but Dyer made sure of the points when he stooped to convert Routledge's 79th-minute center from the left.
Bony applied gloss to the scoreline four minutes later, Hernandez again the provider as the Dutchman out jumped Turner.
The result means Cardiff remains second bottom and three points from safety, while Swansea leaps up to 10th in the table.
- Sports & Recreation
- Wilfried Bony
- Nathan Dyer
- Craig Bellamy