The League Two side, currently 64 places lower than the Gunners, triumphed 3-2 on penalties after the match ended 1-1 following extra time at Valley Parade. Only one Bradford player commanded a transfer fee - James Hanson, who cost £7,500 from non-league Guiseley plus the proceeds of a friendly.
Asked if his players felt embarrassed, 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."
He added: "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.
"They defended very well, very deep in their own half, the pitch was a bit slippery and there was no space," Wenger said. "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."
Bradford boss Phil Parkinson said his players - who won their ninth successive shoot-out - should be given credit for out-performing a full-strength Arsenal side.
He said: "I don't think the enormity of our achievement has quite sunk in. When they get up and see the headlines in the newspapers they will realise what they have done. There is a big gulf between the two teams in terms of finances and places in the divisions.
"Arsenal have played their best team and we were three minutes away from beating them, then we have had the setback of conceding a goal and had to dig deep again in extra time. I hope this can be the first of many nights like this for Bradford City. We want to put Bradford on the map again. I want them to have a team to be proud of."