The home side took the lead when Garry Thompson turned a free-kick home after 16 minutes, but Thomas Vermaelen levelled for the north Londoners in the 88th minute to force the tie to extra-time.
Arsenal dominated for most of the night at Valley Parade, mustering a total of 28 shots and 12 on target during the 120 minutes, but were unable to find a winner.
And City eventually triumphed on penalties after Gunners trio Santi Cazorla, Marouane Chamakh and Vermaelen missed their spot-kicks.
“Congratulations to Bradford,” Wenger told Sky Sports. “They had a good start and after that defended well.
“It happened on penalties but overall we had many chances, their keeper had a great game and we couldn’t convert our chances.
“I cannot fault the effort of our team. I believe we gave absolutely everything for 120 minutes and you have to give credit to Bradford on the night; that’s part of the game.
“I don’t think it was because of a lack of quality; we couldn’t convert our chances for different reasons and overall it was real cup game.
“We’ll get over this. It’s part of our job, that’s what we’ll do. Now we’ll focus on the next game. [But] when you go out it is never good enough for Arsenal.”
Bradford City boss Phil Parkinson was thrilled with his side’s performance as they booked their place in the semi-finals.
“The lads were terrific tonight, the discipline was first class against such a talented team like Arsenal,” he said.
“The talk will be about the penalties but the overall performance was excellent tonight. The way the lads stuck at the task was a credit to them.
“The stadium was full, the atmosphere was incredible and we want to put the club back on the map again, these fans deserve it, the city deserves it.”