Despite playing almost an hour with five attackers, Arsenal failed to break down the League 2 club who went on to claim Tuesday's match 3-2 on penalties after a stubborn defensive display.
Wenger started German international striker Lukas Podolski alongside Gervinho and finished the match with attack-minded players Alex Oxlade-Chamberlain, Marouane Chamakh and Tomas Rosicky on the pitch, but it was left to captain and centre back Thomas Vermaelen to send the match into extra time when he equalised in the 88th minute.
Gervinho squandered a glorious chance in front of goal and Francis Coquelin hit the woodwork in the 38th minute. The end-of-match statistics showed Arsenal had registered 28 shots with 12 on target compared to Bradford's five shots with three on target.
"We played with a very offensive team. What is disappointing is [we played] basically over an hour with five strikers and couldn't score," Wenger told his club's website.
"We created a lot of different situations. You have to say they defended very well. It's difficult to play this kind of game.
"I know people will say: 'it's League 2', but a cup game is a cup game. In football you always have a chance if you give everything."
Asked whether he was embarrassed by the defeat, Wenger said: "You feel embarrassed when you don't give everything. I feel the team did fight and will be more disappointed and frustrated.
"I cannot fault the effort. We have put the effort in [and] have given absolutely everything until the last minute. It was a typical English cup game and Bradford got on top of us in the end. We missed three penalties - that's a lot to take."
Arsenal have not won a major trophy since the FA Cup in 2005 and the latest setback in a disappointing season could trigger another departure at the club with Britain's Telegraph newspaper reporting on Wednesday that Manchester United have their eye on Theo Walcott.
Walcott, who did not play against Bradford, has yet to commit to Arsenal after contract talks broke down in August and the 23-year-old becomes a free agent at the end of the season when his current deal expires.
Striker Robin Van Persie went to Old Trafford at the start of the season. Arsenal have felt his loss with the Dutchman so far netting 11 goals in the Premier League for his new team.
Former Arsenal player Thierry Henry has been linked with another loan return to Emirates Stadium but Wenger said before the match with Bradford that there was no news about his compatriot.
