However, Birmingham were level just after the restart through Garry 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.
Wenger accepted his side - whose only other failure to win at home in the Premier League came against Manchester United, which ended 2-2 - were below par.
"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."
Wenger added: "Every game is important so it's hard to say which is the crucial month, but today we lost two points we shouldn't have lost.
"I'm confident the team will re-focus because they are very disappointed. The focus today wasn't good enough."