The Gunners were defeated 3-2 on penalties after three players missed their spot kicks, and the Frenchman blamed his side's inability to convert their chances for their defeat.
Wenger refused, however, to question the application of his team despite their humbling at the hands of the League Two club.
"It's disappointing but congratulations to Bradford, they defended well, started stronger than us," Wenger told reporters. "The second half and extra time was all us but we couldn't convert our chances and Bradford defended very well.
"It was a typical cup game on a difficult pitch, but a playable pitch and overall we lost on penalties. But I cannot fault the effort of the team, we've given absolutely everything until the last minute and it was a typical English cup game.
"Bradford in the end got on top of us and of course we missed three penalties. That is a lot to take.
"When it goes to penalties you know that the worst can happen, you can lose, it's a little bit of a toss-up. When you miss the first penalty of course it's difficult."
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But the 63-year-old insisted that he could not ask for much more from his players and felt the tie was bound to pose a number of challenges.
"I feel we gave a lot," he added. "It's difficult to make the game the whole time, we played with a very offensive team.
"It's disappointing as we basically played for over an hour with five strikers and didn't score, but we created a lot of different situations. Honestly we have to say they defended very, very well.
"It's difficult to play this kind of game and I know that people will say it's division four but a cup game is a cup game and in football there is always a chance if you give everything and they did that tonight, and we put ourselves in a difficult situation by going 1-0 down."
And the under-pressure boss was adamant that he did not feel embarassed by Arsenal's unexpected exit from the competition.
"You feel embarrassed when you don't give everything," he reflected. "I feel the team did fight, and they will be more disappointed and frustrated."