The Blues, who last weekend saw Andre Villas-Boas become the sixth manager to be sacked by Roman Abramovich since the Russian bilionaire bought Chelsea in 2003, had to wait until the second half at St Andrew's to take the lead against second-tier Birmingham.
But two goals in six minutes approaching the hour mark from Juan Mata and Raul Meireles gave Roberto di Matteo a winning start to his reign as Chelsea's caretaker boss.
The Londoners, held to a 1-1 draw by Birmingham in the original tie, could then afford to see Mata's penalty saved by goalkeeper Colin Doyle.
Chelsea will be at home to another Championship team in Leicester City in next week's quarter-finals.
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