The Blues moved five points clear of champions Manchester United, despite failing to score for the first time in 34 games. United can narrow the gap at Hull City on Sunday afternoon.
Birmingham goalkeeper Joe Hart produced a string of fine saves to frustrate the visitors, extending his side's unbeaten league run to 10 and leaving Chelsea with just one victory from their last seven matches in all competitions.
"It's not a good result for us considering what we did on the pitch because Chelsea played well, had a lot of chances, had a lot of heart and the keeper for Birmingham made some fantastic saves," Chelsea manager Carlo Ancelotti told the club's official website.
Florent Malouda mis-kicked in front of a gaping goal as Chelsea continued to dominate after the break and the Frenchman's luck did not improve as he was dismissed for a second bookable offence late on.
Birmingham had the ball in the net in the first half but Christian Benitez was ruled offside when he tapped in from close range. Television replays suggested he was being played on by a grounded Didier Drogba.
Meanwhile, Roberto Mancini began his reign as Manchester City manager with a 2-0 home victory over Stoke City and there was good news for another of the Premier League's Italian coaches when Gianfranco Zola's West Ham United climbed out of the bottom three with a 2-0 home win over Portsmouth.
Liverpool moved up to seventh with a 2-0 win at home to 10-man Wolverhampton Wanderers, while Wigan Athletic against Blackburn Rovers ended 1-1 as did Burnley's home match with Bolton Wanderers.
"For me, today is a good match. The players were fantastic because Stoke are a hard team," Mancini told the BBC.
Rafa Benitez's Liverpool edged to a much-needed 2-0 win over Wolves in a dismal affair at Anfield with goals from captain Steven Gerrard and Yossi Benayoun.
Eight minutes later the hosts, including Alberto Aquilani making his first Premier League start, added a second through midfielder Benayoun's deflected strike.
