Manchester City hasn't given up on winning silverware this year, either, after reviving its faint Premier League title hopes with a 2-0 win over Chelsea. The champions stayed 12 points behind Manchester United.
While City needed second-half goals by Yaya Toure and Carlos Tevez to wear down the European champions, Swansea's victory at Wembley Stadium was never really in doubt from the moment Nathan Dyer scored the 16th-minute opener.
Dyer and Jonathan de Guzman both finished with two goals and Michu grabbed the other in the largest winning margin in the final of English football's second-tier knockout competition. Swansea will play in the Europa League next season.
"The first trophy is always special and next year in Europe will be a nice experience," said Swansea manager Michael Laudrup, the Denmark great.
Bradford was looking to add the Welsh club to a list of Premier League conquests that already included Arsenal, Aston Villa and Wigan in its unprecedented run to the final.
However, the team currently ranked 79th in English football's league pyramid was blown away by Swansea's renowned passing game, with goalkeeper Matt Duke's straight red card for tripping De Guzman in the 56th summing up their day.
"The early goal killed us," Duke said. "It is disappointing but this is a massive plus for the football club, a massive achievement and we have to remember that."
Bradford was the first fourth-tier team to reach a major final since 1962, when the competition was in its second year and the country's major teams didn't take it seriously.
City produced one of its most assured and convincing displays of the season to issue a reminder to United that the fight for the title remains on.
"We can't think it's finished with 11 games to go," City manager Roberto Mancini said. "It's not my mentality or the mentality of the players."
Toure opened the scoring with a classy finish in the 63rd minute, and Tevez came off the bench to smash in the second in the 85th.
Outclassed for the most part, it could have been so different for Chelsea had Frank Lampard's low penalty not been saved by Joe Hart in the 52nd minute after the goalkeeper brought down Demba Ba in one of the visitors' rare attacks.
After missing a chance to move a point behind City, Chelsea could be overtaken on Monday by Tottenham, which visits West Ham. Arsenal is only two points behind Chelsea in the race for the final Champions League spots.
"When you lose you have to be disappointed and this game was important for us because we could have been closer to Man City," Chelsea manager Rafael Benitez said. "But we cannot give up, we have to carry on, try to win our games and get close."
France internationals Moussa Sissoko and Yohan Cabaye capped Newcastle's self-designated "French Day" by scoring goals in a 4-2 win over Southampton in the other league match, lifting the team clear of relegation trouble.
Home fans donned berets and black-and-white striped T-shirts in celebration of the club's 10-strong French contingent, seven of whom were in the match-day squad on Sunday, but it was one of Southampton's French players — Morgan Schneiderlin — who scored the opening goal.
Sissoko equalized and Papiss Cisse gave Newcastle the lead with a dipping volley, only for Rickie Lambert to level for Southampton with his 12th goal of the campaign.
Cabaye's penalty and Jos Hooiveld's own goal clinched a victory which puts Newcastle six points out of the drop zone and condemned Southampton to only a second loss in 10 matches.
"You look at the Premier League table and we have now put ourselves in a position to attack 10th, and that was the importance of the victory today," Newcastle manager Alan Pardew said.