There were no goals either at Craven Cottage where Tottenham Hotspur earned a draw with Fulham but West Ham United eased out of the bottom three with a 2-0 win over Portsmouth.
Chelsea moved five points clear of champions Manchester United despite failing to score for the first time in 34 league games.
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 five.
Custodian Hart, who ended the match with a bandaged head, denied Daniel Sturridge and Frank Lampard in the first half.
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