The Gunners had gone into the day at the top of the Barclays Premier League, but failed to secure victory in front of their own fans as Alex McLeish's men produced a battling display to earn a 1-1 draw.
"It was very frustrating," said the Arsenal boss. "The crowd pushed us on, but we were punished for losing concentration on the set-piece. We thought it was going to be too easy, but we didn't create enough overall. We were not sharp enough and they defended well. It's as simple as that."
Everything looked to be going to plan when Emmanuel Adebayor's penalty put Arsenal ahead after 21 minutes after Stephen Kelly was adjudged to have brought down Eduardo from behind.
However, Birmingham were level just after the restart through Gary O'Connor's deflected header following a corner, and the visitors held on under immense late pressure to boost their own confidence of staying in the top flight.
Arsenal were missing Robin van Persie and Tomas Rosicky, both of whom had a virus, while Kolo Toure, Emmanuel Eboue and Alex Song are all away at the African Nations Cup.
Wenger, though, refuted suggestions his squad was in need of reinforcements, saying: "I don't look for anybody. The last excuse you should look for when a team that has won many games this season doesn't win is to say you should buy players."
Blues manager McLeish, meanwhile, was full of praise for his battling side, who were lead by captain Damien Johnson - playing with a protective mask two weeks after suffering a slight skull fracture.
"We were playing against a special team and the determination of the players got us through," said the former Scotland manager.
"Arsenal can go through you like a hot knife through butter - but we had to concentrate for 90 minutes. We broke them up a few times and defended very well. They sniffed the danger and snuffed out the dangers most of the time."
