Having earlier spared the Gunners from humiliation with a late equaliser, captain Thomas Vermaelen went from hero to villain as he hit the post with a penalty which would have taken the shoot-out to sudden-death.
With so many high-placed Premier League teams already out, this competition represented Arsenal's best chance for a trophy but, barring an FA Cup victory, their trophyless run will almost certainly extend to an eighth season.
For most of the match it looked as though Arsenal would be eliminated in 90 minutes before Vermaelen rescued them with an 87th-minute header to ensure extra-time.
It was not so long ago that Arsenal's trophy ambitions were ended by the likes of Barcelona and Lionel Messi, but Bradford and their goalkeeper Matt Duke have now added their names to the Gunners' tale of woe.
Yet as Arsenal found themselves pinned back on the ropes by Bradford in the early stages, perhaps being more accustomed to more glamorous surroundings was the problem.
A bobbly pitch, a raucous home crowd and a collection of unheralded players determined to test the nerve of Arsenal's stars clearly unsettled Wenger's team.
Bradford's opening goal, on 16 minutes, came as a result of a clumsy challenge by Vermaelen and Arsenal couldn't deal with Gary Jones' free-kick before Thompson volleyed the ball into the net from a tight angle.
Arsenal had a glorious chance to equalise as Gervinho somehow contrived to miss an open goal from four yards just before the break.
Rather than a second-half onslaught, Bradford were reasonably comfortable but just when Arsenal needed a hero, Vermaelen rescued them with his dramatic late equaliser.
However, after failing to score in extra-time, Arsenal paid the penalty.