Blues' victory was only their second in nine league games while it ended a three-match winning run for the in-form Clarets.
Danny Ings twice headed off target from Ross Wallace crosses before City broke the deadlock after 21 minutes thanks to Curtis Davies, just as he had done in a 2-2 draw between the two sides just before Christmas.
It came courtesy of a familiar double act for Blues fans, on-loan West Ham winger Robert Hall delivering a pin-point corner which Davies met from close range to convert.
Ings, Burnley's liveliest outlet, ratted the underside of the crossbar just four minutes after the restart.
Paul Robinson was guilty of bringing down Junior Stanislas in the area and Wallace smashed the resulting penalty down the middle to level matters.
Davies headed straight at Lee Grant before Burnley were then reduced to 10 men, Dean Marney dismissed for diving in on Robinson in the middle of the park.
Marlon King had the final say in the 90th minute when he fired home from Nikola Zigic's header.