The Premier League Latics side, featuring nine changes from Saturday's victory over West Ham, created a host of opportunities but could not make a breakthrough during the 90 minutes or in extra-time as their spirited League Two opponents held on for spot-kicks.
Shaun Maloney then blazed his penalty over and Jordi Gomez saw his saved as the visitors advanced to the last eight of the competition.
Wigan looked comfortable in the early stages and thought they had taken the lead in the 10th minute when Ivan Ramis, captain for the night, headed home Gomez's free-kick, only to see the referee's assistant raise a flag.
Five minutes later Mauro Boselli found himself through on goal, but he made a hash of an attempt to go around Matt Duke, appearing to roll his foot over the ball, and the goalkeeper was able to save.
Bradford struggled to get into the game, but after Nahki Wells almost pounced as Ali Al Habsi rushed to clear away a poor back-pass by Ramis, the visitors seemed to grow in confidence.
Wells struck an effort off-target moments later and the Bantams then had a penalty appeal waved away as James Hanson went down in the box under the attention of Ben Watson.
Wigan almost grabbed the lead right at the end of the first half when Gomez met a lofted ball into the area first time, bringing a good stop out of Matt Duke.
The hosts looked to take control of proceedings again after the break, with David Jones having a shot saved before Gomez just failed to finish from Boselli's tee-up as Bradford scrambled the ball away.
The introduction of Maloney and Jean Beausejour for Callum McManaman and Redmond gave the Latics more impetus, with Fyvie dragging a shot wide and Boselli striking one low and hard which Duke got behind.
Bradford skipper Nathan Doyle then sent an effort skidding just the wrong side of Al Habsi's post and, after Boselli miscued a header in front of goal and Maloney fired wide, the contest went into extra-time.
Wigan piled on the pressure at the start of the additional period, with Duke doing well to deny Boselli, who also hit the side-netting in between a Beausejour attempt that deflected over and Gomez shot that went wide.
Boselli again failed to finish shortly into the second half of extra-time, clipping tamely into Duke's arms, and Gomez was then guilty of profligacy, fluffing a one-on-one opportunity.
Bradford were unerring with their spot-kicks, but Maloney ballooned his over before Duke blocked Gomez's attempt to send Wigan crashing out and the visitors through.