First half goals from Nathan Dyer and Michu set the Swans on their way before Dyer grabbed a second early in the second half to make it 3-0.
Bradford keeper Matt Duke was then shown a red card in the 56th minute after he had brought down Jonathan de Guzman who picked himself up to convert past Jon McLaughlin before adding his second in injury-time to complete the five-goal rout.
Swansea will now move into the Europa League next season just ten years after the club were almost relegated from the Football League when they were languishing in the fourth tier.
The 5-0 scoreline was also a record winning margin for the League Cup giving Swansea a trophy in their centenary celebration season.
2012/13 Capital One Cup final
Wembley Stadium, London - 82,597