The winger netted in the 16th and 47th minutes before standing over the ball when Bantams keeper Matt Duke was sent off for bringing down Jonathan de Guzman in the area.
At the spot, Dyer was pushed aside by de Guzman despite him already having scored twice which would have given him the opportunity to become the first man in history to score a treble in the league cup final. De Guzman netted the ensuing spot-kick.
"To score a hat-trick in an historic moment like this... I thought I should have got the ball, but it wasn't meant to be, and I passed it on and that's the way it went,” Dyer told Sky Sports 1.
Dyer added of the win: "Swansea have come so far as a club and a city. For us to win this cup final is massive for us, the player have worked so hard all week, and it's just a great day."