Following a tentative start at Stamford Bridge, Chris Hughton's visitors broke the deadlock in the 20th minute. Nathan Redmond's free kick flashed across the Chelsea goal and full-back David Murphy was on hand to turn the ball in from close range for Birmingham.
However, Chelsea had a chance to level up proceedings just one minute later when Wade Elliott was adjudged to have brought down Ramires inside the area. Juan Mata took the resultant spot kick but Birmingham goalkeeper Colin Doyle guessed correctly to turn the Spaniard's effort wide of goal.
Daniel Sturridge went closest for Chelsea before the break, but his long-range effort grazed the crossbar.
The half-time introduction of Didier Drogba in place of Fernando Torres gave Chelsea more of an attacking threat after the break as the home side began to dominate.
That pressure told just after the hour mark, when Sturridge headed a cross from Branislav Ivanovic beyond the reach of Doyle.
Both teams had chances to advance to the quarter-finals in the remaining minutes, as Birmingham midfielder Jordan Mutch saw his driven free kick held by Petr Cech, while Ivanovic lifted his effort over for Chelsea from close range.
The two teams will meet again at St Andrews in around ten days' time to determine who will progress through to the last eight.