The south Wales outfit cruised to a record 5-0 win over League Two Bradford, collecting the first major silverware in their 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."
It completed a hat-trick of Wembley wins for Laudrup, who was also successful at the stadium with Denmark against England in 1983 and 1992.
"Wembley is one of the special stadiums in world football," he said.
"I have some very good memories from 1983 and 1992. Could we play some more games here?"
Laudrup refused to answer questions about his own future, although he admitted the victory, in Swansea's centenary year, compared with any in an illustrious career.
"It is one thing to win a trophy with Barcelona, Real Madrid or Juventus, but to win one with a smaller team like Swansea is absolutely fantastic," he said.
"It is their first major trophy ever and, especially in this season, is up there with the best things."
And it came just a decade after Swansea almost got relegated to the Conference.
"This club nearly went out of League Two," he said.
"You are talking about your life.
"Two years ago they played to go from the Championship to the Premier League.
"That is a world of difference as well.
"Maybe those two things are most important. But the first major trophy ranks highly in the history of any club."
Bradford manager Phil Parkinson refused to look on the bleak side, praising his team's supporters, most of whom stayed for the trophy presentation.
"I would have loved to make more of a game of it but it was a tough afternoon," he said.
"One of the Swansea players' wages probably covers our budget for the whole season. That is the gulf we were dealing with.
"The lads are a bit down in the dressing room because we are a better team than we showed but Swansea's all-round movement was very difficult for us to cope with on a big Wembley pitch."
Swansea formed a guard of honour at the bottom of the Wembley steps to applaud Bradford for their achievement in becoming the first tier four side to reach a major final at the world-famous stadium.
But Parkinson did leave with a sense of regret that Friend had not opted to keep Duke on the pitch rather than condemning him to become the first keeper to be sent off in a League Cup final.
"The referee could have used his common-sense in the context of the game," he said.
"We were 3-0 down against a Premier League team and a penalty had just been awarded against us.
"I don't think we were going to come back to win 5-4.
"He could have used a bit of discretion and just give Matt a yellow card and that would have sufficed."