The home side took the lead when Garry Thompson volleyed in but Thomas Vermaelen headed in two minutes from time to send the match into extra-time and penalties.
Bradford twice failed from the spot but Santi Cazorla, Marouane Chamakh and Vermaelen were equally culpable to hand the League Two side a 3-2 success on penalties and a place in the semi-finals for the first time in their history.
"Congratulations to Bradford, they had a good start and after that they defended well," Wenger said.
"Overall we had many chances. The keeper had a great game and we couldn't convert our chances.
"When you go out it's never good enough at Arsenal. I think we lost on penalties to a team who defended well.
"I cannot fault the effort we gave for 120 minutes. You have to give credit to Bradford on the night. That's part of the game.
"You feel embarrassed when you don't give everything, I think (the players'll) be more disappointed and frustrated.
"It's not a lack of quality. We couldn't convert our chances for different reasons. Overall it was a real cup game.
"We will get over that. It's part of our club, it's what we do."
Wenger felt the surface at Valley Parade did not help the visitors.
"It was a typical English cup game on a difficult pitch and overall we lost on penalties but I cannot fault the efforts, we kept going until the last minute and in the end Bradford got on top of us. The pitch was a bit slippery and there was no space.
"Of course we missed three penalties and that's a lot to take."
The Gunners did not force a save out of Bantams keeper Matt Duke until the 69th minute but Wenger insisted his team had created enough overall.
"We played with a very offensive team and for over an hour played with five strikers but couldn't score.
"Where does it leave our season? To focus on the next game, sport is about that."