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.
Bradford, whose huge contingent of flag-waving fans made a memorable impression at Wembley, never seriously threatened to score in the very one-sided final having previously dumped out top flight Wigan Athletic, Arsenal and Aston Villa.
Welsh side Swansea, who have only been in the Premier League for two seasons, took the lead when Dyer slid the ball home in the 16th minute and doubled it five minutes before the break when Michu scored with a cool finish.
Dyer ended the match as a contest with a superb curling shot soon after the restart.
De Guzman made no mistake with his 59th-minute penalty after winning a heated row with hat-trick chasing Dyer over who would take the spot-kick. The midfielder added another in stoppage time.
"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."
Swansea dominated from the opening whistle while Bradford, who showed so much enterprise and verve in knocking out Premier League opponents on their stunning run to the final, sat back and defended allowing the top-flight side to take the initiative they never relinquished.
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.
The game then developed a familiar pattern of Swansea attacking almost at will and Bradford defending with only the sporadic foray upfield and it was no surprise when Swansea made it 2-0 in the 40th minute.
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.
Dyer made it 3-0 with a wonderful goal after a one-two with Wayne Routledge before De Guzman grabbed his brace.
"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
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