The Welsh outfit entered Sunday's showdown with League Two giant-killers Bradford City at Wembley Stadium as overwhelming favourites, but aware that an upset could be on the cards.
In the end, the Swans swept to a convincing 5-0 success, with Nathan Dyer and Jonathan de Guzman netting a brace apiece and Michu grabbing the 19th goal of a remarkable campaign for him on a personal level.
Laudrup was full of praise for his side at the final whistle, with the Dane having made history in just his first season in charge at the Liberty Stadium.
Asked how he felt as the celebrations started around him, Laudrup told Sky Sports: "Very proud.
"To do what they have done on their way to Wembley, Bradford, they have beaten Arsenal and Aston Villa, among others. It's not easy but after the second goal you could see the difference.
"I am very proud of my team today.
"They have never been behind against a Premier League team. We had to move the ball with good pace and score the first goal. We did that. I think we played a very good game. We were great."
Pressed on where the achievement ranks on his personal honours board, the former Barcelona and Real Madrid midfielder added: "As a manager, it is absolutely at the top. Winning a trophy for the first time in 100 years...
"The first trophy is always special and next year we will be in Europe, it will be a nice experience."