Two goals each from Nathan Dyer and Jonathan de Guzman, one from the penalty spot, and another from Michu saw the Barclays Premier League side clinch the first major trophy in their 100-year history.
Laudrup told Sky Sports 1: "I'm very proud of my team today [Sunday]. I think it was a great performance."
The 48-year-old Dane, who replaced Brendan Rodgers last summer, also hailed the triumph as the best of his managerial career.
He said: "As a manager it's absolutely at the top, winning a trophy for the first time in 100 years.
"The first trophy is always special, and now next year in Europe... it's a nice experience."
Fourth-tier Bradford had enjoyed a fairytale journey to the final, including knocking out top-flight opposition in the form of Arsenal, Aston Villa and Wigan, and Laudrup felt getting the first goal would be crucial for his side.
He said: "They had never been behind against Premier League teams, so we had to move the ball at good pace and we have to score the first goal.
"And after getting second goal it was a little easier.
"I think we played a very good game and moved the ball around, I think we did great.
"When you are 3-0 up and with confidence like that you see the difference from a Premier League side and League Two but I have to say what Bradford have done this season is impressive, although it is also impressive what we have done."
Swansea captain Ashley Williams, meanwhile, hailed the fulfilling of a lifetime ambition.
"This is what we all dreamed of as little boys. It's a dream come true winning the final at Wembley," he said.
"We had a job to do and did it brilliantly. We knew there was a danger as they have beaten Premier League teams, but we had a job to do.
"The boys were really professional and we carried on playing all the way."
Dyer opened the scoring in the 16th minute and the prolific Michu doubled Swansea's lead five minutes before half-time.
Dyer then notched his second two minutes into the second half to make it 3-0 and he was eyeing a record-breaking hat-trick in the 59th minute when Swansea were awarded a penalty.
However, De Guzman refused to stand down as penalty taker, much to Dyer's obvious anger as he unsuccessfully tried to persuade his team-mate to give him the opportunity to become the first man in history to score a hat-trick in a League Cup final.
Of that incident, Dyer said: "To score a hat-trick in an historic moment like this... I thought I should have got the ball, but it wasn't meant to be, and I passed it on and that's the way it went."
Dyer added of the win: "Swansea have come so far as a club and a city. For us to win this cup final is massive for us, the player have worked so hard all week, and it's just a great day."