The League Two Bantams were three minutes away from a normal-time win at Valley Parade, with Thomas Vermaelen`s equaliser cancelling out Garry Thompson`s early opener.
But City were not thrown off course and hung on for a penalty shoot-out which they duly won 3-2 - their ninth successive shoot-out win.
Wenger's decision to play many first team players was a sign that he was determined to end the seven-and-a-half year trophy drought they currently going through.
Despite the loss, Wenger believes that he side did not underperform and gave credit to the Bradford side.
"Congratulations to Bradford, they had a good start and after that they defended well," he said.
"Overall we had many chances. The keeper had a great game and we couldn`t convert our chances."
When probed on the magnitude of the Gunners losing to a League Two side, Wenger added: "When you go out it`s never good enough at Arsenal."
When asked if it was a humiliating loss, he said: "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.
"It`s not a lack of quality. We couldn`t convert our chances for different reasons. Overall it was a real cup game."
He concluded by saying: "We will get over that. It`s part of our club, it`s what we do.
Bradford manager Phil Parkinson hopes his club are on the way back after they stunned Arsenal in the quarter-finals of the Capital One Cup.
It is a decade since Bradford were in the Premier League but now, with a semi-final place secured, Parkinson hopes they are on their way back to the top.
"The lads were terrific, the commitment was first class and considering we were three minutes away.... People will talk about penalties but our overall performance was outstanding," he told Sky Sports 1.
"Arsenal have got world class players and played their strongest team, but you have to give the players so much credit.
"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."
Midfielder Gary Jones, a scorer in the shoot-out, called it "one of the best nights of my life" before adding: "We`ve just knocked Arsenal out of the Capital One Cup. It was 11 v 11 and we had to keep our shape and frustrate them."
Goalkeeper Matt Duke saved a penalty from Santi Cazorla and said: "We set our stall out and the men in front of me were fantastic. We are really fit as a team and that showed."