The Bantams triumphed 3-2 on penalties in the Capital One Cup to reach a first ever semi-final after the match finished 1-1 after extra-time.
The spot-kick specialists won their ninth successive penalty shoot-out 3-2 after Thomas Vermaelen struck the post. Vermaelen had earlier appeared to rescue the Gunners with a header three minutes from normal time to equalise Garry Thompson's first-half volley.
Wenger said: "You feel embarrassed when you don't give everything but I feel the team did fight and they will be more disappointed and frustrated.
"It's disappointing but congratulations to Bradford. They defended well, started stronger than us and in the second half and extra time it was all us but we couldn't convert our chances.
"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.
"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.
He added: "They defended very well, very deep in their own half, the pitch was a bit slippery and there was no space. 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."