After neither side were able to score in the 90 minutes, Wednesday missed three of their four penalties while Bradford scored all three through Ritchie Jones, Michael Flynn and substitute Ross Hannah.
Bradford striker James Hanson missed the best chance of the first 45 minutes by heading wide after Mark Stewart's cross was deflected into his path.
However, Bradford, watched from the stands by new manager Phil Parkinson, dominated the second half against an under-strength Wednesday side weakened still further when manager Gary Megson substituted three first-team squad players - keeper Nicky Weaver in the second minute and David Prutton and Jose Vitor Sarmego in the 18th minute.
Stewart created a good chance for himself in the 49th minute, taking the ball from defender Ayo Obileye after chasing a long pass only to see substitute keeper Richard O'Donnell block his shot.
Defender Luke Oliver also came close to scoring in the 65th minute when his header was cleared from under the crossbar by substitute Cecil Nyoni with O'Donnell beaten. Then, Jones' volley was saved on the line by O'Donnell in the 83rd minute before Robbie Threlfall's free-kick was headed against the crossbar by Oliver in stoppage time.