Man-of-the-match Nathan Dyer opened the scoring in the 16th minute at Wembley, before Michu’s 19th goal of the season put the Swans 2-0 up.
Dyer then added a superb second after the break, before Jonathan de Guzman’s penalty and late strike added gloss to the scoreline after Bradford had goalkeeper Matt Duke sent off.
“As a player I had the luck to play in the biggest teams in different countries,” Laudrup told BBC Sport after the win, which confirms Swansea’s place in next season’s Europa League.
“Swansea is not the biggest team in the Premier League but we are here winning a trophy and that is fantastic. To win a trophy with a smaller club is outstanding.
“It’s a great performance and I’m very proud of my team today. We said we had to move the ball with a good pace and score the first goal and we did that. The first trophy is always special and next year in Europe will be a nice experience.”
Michu, who has enjoyed a stunning first season in England after joining Swansea from Rayo Vallecano last summer, was thrilled to collect his first winners’ medal with the club.
“It was a great performance by the team and I think we deserved this,” he said.
“It was good for me too to score a goal. It is difficult to score five goals but we did it and we deserve to win this final. It is a really good season, my first here. We are in a good position in the league and to win this medal is very good.”
Meanwhile, Bradford boss Phil Parkinson preferred to look ahead to the rest of the season rather than dwell on the disappointing defeat.
“We will dust ourselves down,” he said. “We have enjoyed the whole experience and we will go back up to Bradford and enjoy the last 15 league games. We have a chance of promotion so we will make a go of it.”