"Everyone is devastated in the dressing room, they (the players) are destroyed, and I don't want to add to that," the Arsenal manager told reporters after Birmingham had beaten his side 2-1 at Wembley.
"I am bitterly disappointed like the whole team. We had some problems to start the game, the number of games we played caught up a little bit on us."
The North London club, whose last silverware was the 2005 FA Cup, again attracted criticism that they lack the mental strength and physical power to turn talent into trophies.
Birmingham's winner came in the 89th minute after a defensive blunder by keeper Wojciech Szczesny and defender Laurent Koscielny, leaving Arsenal players slumped on the turf at the final whistle.
Asked if the defeat could affect the rest of Arsenal's season, Wenger said: "It can. We are tremendously disappointed to lose, and we face a lot of questions after that mistake.
"We have to be strong enough to stand up to it and show that we have the mental strength to respond.
"That's what a team is about; these kind of things are part of the game. We had enough chances to kill off the game when it was 1-1, but we didn't.
"What we have achieved so far is amazing - only one defeat in 18 games, and in the last minute here."
POTENTIALLY DECISIVE
Arsenal face a potentially decisive March.
They play third-tier club Leyton Orient in an FA Cup Fifth ound replay at Emirates Stadium on Wednesday and, if they progress, a quarter-final tie at Manchester United.
On March 8 they travel to Barcelona hoping to defend a 2-1 lead in the second leg of their last-16 Champions League tie and they will continue the quest to overhaul United's four-point lead at the top of the Premier League.
"We have to pick ourselves up because the challenges ahead are massive," Wenger said. "We are in a good position."
The Frenchman fears more injuries could scupper his team's hopes.
They were without Spain midfielder Cesc Fabregas and England winger Theo Walcott on Sunday, and Dutch striker Robin van Persie was forced off against Birmingham.
"Van Persie has a knee injury and we will assess it (on Monday)," Wenger said. "But we cannot afford to lose players to injuries any more."
