Phil Parkinson's spot-kick specialists won their ninth successive shoot-out after Thomas Vermaelen struck the post. The Arsenal skipper had earlier appeared to rescue the Gunners with a header three minutes from normal time to equalise Garry Thompson's first-half volley.
But keeper Matt Duke defied the Premier League side to send the tie to a shoot-out - and glory for Bradford.
Hope turned to jubilation for the Bradford fans in the 16th minute when a free-kick led to Thompson's goal. Vermaelen had blocked off Wells and Gary Jones caused mayhem to Arsenal's zonal markers with his free-kick, Will Atkinson flicking it on for an unmarked Thompson to gleefully fire home.
It could have been 2-0 after Jones' corner found Rory McArdle and only Kieran Gibbs' clearance on the goal-line prevented further embarrassment for Wenger's men. Arsenal needed inspiration from somewhere and Francis Coquelin was almost the unlikely source with a skilful dribble through two challenges before rattling a shot against the foot of the post with Duke looking beaten.
Bradford's goal took on a charmed life when a more typical Arsenal move ended with Gibbs driving in a cross and Gervinho, from two yards out, somehow contriving to let the ball go through his legs.
It was well past the hour when Duke was forced into his first save, parrying Alex Oxlade-Chamberlain's first-time shot from fellow substitute Tomas Rosicky.
Two brilliant saves in quick succession by Duke from Cazorla took the game into extra time. Finally, Arsenal started to look like the Premier League side as Rosicky and Wilshere started to pull the strings. Duke's stunning save denied Gibbs, then Sagna's header was cleared off the line by Jones.
Cazorla nearly won it with a stunning, swerving effort from fully 30 yards that struck the crossbar, but Bradford held out for the shoot-out.
Arsenal were always on the back foot from when Cazorla had their first spot-kick saved, even more so when Marouane Chamakh hit the post. Bradford could even afford to see Wojciech Szczesny deny Ritchie Jones and Stephen Darby before Vermaelen's miss sent Valley Parade into ecstasy.