Jose Mourinho's side extended their advantage to 10 points over Manchester United, who make the short trip to the Reebok Stadium to face Bolton Wanderers later on Saturday.
Mourinho was unhappy with another unconvincing display away from home and Chelsea have won just one from their last five on their travels.
"I am concerned, I want to win every game, I'm not happy with the performance and result," he told Sky Sports.
"We did not deserve to win the game, Birmingham deserved a point... they showed great spirit. I have to praise the way they fought.
"It's (the title) in our hands. The important thing is we keep winning our matches at home and we will be champions again."
Birmingham, third from bottom and reeling from shipping 10 goals in their last two games in league and Cup, defended competently and produced the game's best moment when Olivier Tebily forced Petr Cech into a fingertip save from long range.
Chelsea almost stole the points at the death but Argentine Hernan Crespo wasted two good chances while defender Ricardo Carvalho headed over.
The leaders also had two goals ruled out, Mourinho lamenting the decision to nullify Asier Del Horno's 50th minute header after Carvalho, in an offside position, was deemed to be active.
"We scored a goal... it was not offside. It was a clear goal according to the rules. I don't want to say more about it because every single word I say in this country the next day is a nightmare."
Third-placed Liverpool play away at West Bromwich Albion in a late kickoff.