Claudio Ranieri's unbeaten Chelsea side had arrived at St Andrews with a game in hand and needing just a draw to lead the table on goal difference.
They achieved that in a dour encounter at St Andrews to take them to 20 points from eight games, with a goal difference of 11 compared to Arsenal's 10.
Manchester United are one point back, also from eight games, with Birmingham moving past Fulham into fourth on 15.
Chelsea will have gained confidence for their next match at Arsenal on Saturday in what has become a top-of-the-table clash.
"It will be a very exciting match. For us it's important to continue this way. I have seen a good performance (by Chelsea) so far," manager Claudio Ranieri told Sky Sports.
His opposite number Steve Bruce added: "It was a fantastic effort by us against a really good Chelsea
Neither goalkeeper was really tested in the first half on Tuesday night although Maik Taylor was the busier as Chelsea, starting with Jimmy Floyd Hasselbaink and Hernan Crespo up front, dominated possession.
Birmingham, however, unbeaten at home this season, remained solid at the back while leaving Christophe Dugarry as a lone threat up front.
Taylor saved them in 64th minute when he tipped a Hasselbaink shot around the post following a through ball from Frank Lampard. The keeper then denied the England midfielder with 11 minutes remaining by tipping his shot over the bar.
"Whey you're as determined as we were in the second half sooner or later you'll find a goal," said Ranieiri. "(Hasselbaink and Crespo) tried to do something but against three, four, five players the space was very small."
Chelsea striker Mikael Forssell, on loan at Birmingham, was not cleared to play in Tuesday's game by the London club.