Phil Parkinson’s Bantams of League Two had knocked out Wigan Athletic, Arsenal and Aston Villa - the latter over two legs in the semi-final - to seal their place in the showpiece League Cup final.
It was an historic Wembley appearance from the men in claret and amber who had defied the odds to become the first team from the fourth tier to reach a major final.
But Swansea - who were playing in their first ever final - had too much class, pace and skill and were emphatic winners.
They completely dominated the Bantams, who perhaps overawed by the occasion, were a shadow of the side that took the fight so famously to Arsenal and Villa in the previous rounds.
Nathan Dyer was a complete menace all day, scoring in each half to cap a fine performance.
The tricky right winger pounced in the 16th minute to guide the ball over the line after City keeper Matt Duke parried Michu’s left-foot shot across goal.
Spanish striker Michu then got in on the act, drilling a low shot past Duke from just inside the area five minutes before half-time.
Parkinson’s Bantams had not mustered a shot on goal and any hope of a recovery in the second half was quickly erased with two goals in the first 11 minutes of the second half.
Dyer cut inside onto his left foot and fired home his second and Swansea’s third on 47 minutes.
Then on 56 minutes, Duke - so often the hero of this Cup run - was sent off for bringing down Jonathan De Guzman as attempted to go around him.
De Guzman dusted himself off and fired home the fourth past reserve goalkeeper Jon McLaughlin.
Ads the game petered out, with Bradford finally having a shot on goal in the 87th minute with a tame volley from Gary Jones, Swansea claimed a fifth in stoppage time when Jonathan De Guzman scored at the near post.