The south Wales outfit cruised to a record 5-0 win over League Two Bradford, collecting the first major silverware in the club's history in addition to booking their place in next season's Europa League.
Indeed, the only sour note for Swansea was the penalty row between Nathan Dyer and Jonathan de Guzman after the latter had been brought down by Bradford keeper Matt Duke inside the box.
As referee Kevin Friend brandished the red card to Duke and Bradford introduced Jon McLaughlin, Dyer attempted to wrest the responsibility away from De Guzman, knowing he was one goal away from becoming the first player to score a hat-trick in a League Cup final.
De Guzman would not back down, leaving Dyer simmering, although the pair did make up once the Dutch midfielder had scored. And Laudrup offered a simple explanation for the remarkable scenes.
"There was no designated penalty taker. I forgot," he said."This was our 36th game of the season and it was our first penalty so it is my fault because I didn't say who was going to take it if we got one. I thought we would go through the whole season without getting one."
As it turned out, De Guzman also scored twice, wrapping up a handsome victory in stoppage time. In addition Michu found the net, splitting Dyer's brace, the wide man putting Swansea in front after 16 minutes and then curling home a superb effort just after the break.
It was part of a Swansea demolition job, with Bradford not managing a corner until the 83rd minute and a shot for two minutes after that as their remarkable story came to a shattering end.
"To lift the trophy is great," said Laudrup. "But the way we did it says a lot about our performance. We all know what Bradford have done this season. It is absolutely outstanding.
"This final will remain in the history books, a small part because of us, a large part because of Bradford. That is why I am so happy with the way we did it."