However, Lee Bowyer capitalised on a mistake by young keeper Vito Mannone - preferred to fit-again Manuel Almunia - to pull a goal back before half-time.
After soaking up plenty of pressure from the battling Blues, substitute Andrey Arshavin wrapped up the points with a well-taken strike five minutes from time to make it seven successive wins in all competitions.
"We were crushing, in control, and wondering who would score third goal," said Wenger, "but we made a mistake defensively and they got a goal back. The Premier League is like that.
"In the second half, we did not have the same fluency, control, or pace. However, 3-1 is still a fair result because we still had plenty of chances. Birmingham worked hard and regrouped well when they did not have the ball.
"We also lacked a bit of sharpness and sometimes tried to force a situation individually when it was possible collectively, and that is why we did not score more goals."
Wenger added: "When we focus to play our game, I believe we can score and play the ball through any space.
"That is something fantastic, but we have to keep our composure."