Two goals from Nathan Dyer, one from Michu and a penalty and a late strike from Jonathan de Guzman ended Bradford's fairytale cup adventure in which they became the first fourth-tier team to reach a big English final for more than half a century.
Bradford goalkeeper Matt Duke was sent off early in the second half for a trip on De Guzman before the Dutchman netted the spot-kick as Swansea went on to record the biggest ever League Cup final win and secure a place in next term's Europa League.
"I'm very proud," Swansea manager Michael Laudrup told Sky Sports after winning a trophy in his first season in charge.
"It wasn't easy but after the second goal you could see the difference. I think we played very well. As a manager it's at the very top to win a trophy for the first time in 100 years."
In essence, to have any chance of winning the game, Bradford had to reach half-time without conceding a goal but that target ended when Dyer reacted quickly to slam in Duke's parried save from a shot from Michu.
Defender Ben Davies began the move with a pass to Pablo Hernandez, who then fed Michu whose left foot shot went through Carl McHugh's legs and past Duke into the far corner for his 19th goal of the season.
"Conceding the goals when we did made it very difficult for us," Bradford boss Phil Parkinson said. "It was tough for us today but enormous credit to our guys for getting to the final."
Swansea, who have won the Welsh Cup 10 times and formerly represented Wales in the old European Cup Winners' Cup, will now be one of England's representatives next season on their return to Europe after an absence of more than 20 years.
"Swansea have come so far as a club and
