Chelsea striker Claudio Pizarro grabbed a late winner as the visitors toiled to victory over a tenacious Birmingham.
Chelsea cannot hide their delight after Pizarro's winner
Birmingham nullified Chelsea for most of the game and saw Cameron Jerome head against the post from a poor clearance by Chelsea keeper Petr Cech.
Birmingham's Sebastian Larsson mistimed a shot from point-blank range, while Nicolas Anelka had a one-on-one effort saved by Maik Taylor at the other end.
But Pizarro headed in the winner as Chelsea won without playing well.
Chelsea had dominated possession in the early stages but struggled to prise a way through a disciplined and determined home outfit.
Birmingham energetically snapped at the heels of their opponents and gave them little sniff of finding a way through their organised ranks.
The St Andrews side were playing more like an away team and Garry O'Connor should have done better for them than to head a corner wide when unmarked.
Olivier Kapo also got in a header for Birmingham but his effort lacked power and Cech gathered comfortably.
Chelsea's cause was not helped by Shaun Wright-Phillips failing to run off an injury after having landed awkwardly when going up for a header with Damien Johnson early on.
The winger was eventually forced off with striker Pizarro coming on before defender Alex glanced a header wide for the visitors.
Chelsea's failure to turn their possession into goals was almost punished by Birmingham as the home side finished the first half strongly.
The hard-working Midlands side pressured Cech into a poor clearance which Jerome instinctively diverted towards goal only to see his header hit the post and go wide.
Birmingham striker Jerome was a constant threat to Chelsea
And Chelsea were hanging on when Jerome had two successive shots cleared from their goalmouth by Alex and Juliano Belletti.
Fabrice Muamba sent a shot from the edge of the Chelsea box over as the visitors' defence was stretched and also thankful when the half-time whistle was blown.
Chelsea's backline had been unnerved by the tough-tackling and closing down of Birmingham and a backpass from Alex almost played O'Connor in on goal only for the alert Cech to slide out and clear the danger.
The home side were now on the front foot and Larsson had a glorious chance to reward their play with a goal.
Kapo pulled the ball back from the byline for the Swede, who mistimed his effort at the near post and saw it loop off target after catching him by surprise.
The fear that the quality of the visitors would see Birmingham regret not making more of their attacks was then almost realised.
Anelka had a clear sight of goal from 12 yards but shot straight at Taylor as he was put under pressure by defender Rafael Schmitz.
Birmingham brought on new signing James McFadden and he immediately got involved as he had an angled shot saved by Cech.
But Pizarro stole in to head in a Belletti corner and clinch Chelsea a victory to keep them in touch with Arsenal and Manchester United at the top of the table.
Birmingham boss Alex McLeish:
"We put in a great performance, we defended well in open play and we worked hard.
"We had a brilliant 15 minutes before half-time when we had lots of attempts on their goal but we didn't have and the post saved Chelsea.
"I am very proud of the players and on another day we would have got something out of the game."
Chelsea manager Avram Grant:
"It was very important to win. It was not high quality but we deserved to win.
"We did not create that many chances. There was one but at the end of the day we won and that's what matters.
"We knew it would be a difficult game against Birmingham. They defended very well and had a high spirit."
Birmingham: Maik Taylor, Kelly, Schmitz, Ridgewell, Queudrue, Larsson, Muamba, Johnson, Kapo, O'Connor (Forssell 72), Jerome (McFadden 72).
Subs Not Used: Doyle, McSheffrey, Parnaby.
Chelsea: Cech, Belletti, Alex, Carvalho, Ashley Cole, Wright-Phillips (Pizarro 29), Makelele, Ballack, Joe Cole (Sidwell 85), Anelka, Malouda (Bridge 90).
Subs Not Used: Cudicini, Ben-Haim.
Goals: Pizarro 79.
Ref: Rob Styles (Hampshire).
BBC Sport Player Rater man of the match: Chelsea's Nicolas Anelka 7.36 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.