After scoring a stunning opener just before the break, Burke added what proved to be the winner just 10 minutes from time.
Billy Sharp scored a stoppage-time penalty for the visitors, but it proved too little too late for Forest.
Marlon King then had another chance for the Blues, but he could only hook his low effort wide of the target.
Butland was forced into a save from point-blank range just before the break, when Greg Halford latched on to Reid's clever free kick.
However, Birmingham took the lead right on the stroke of half time when Burke's stunning long-range effort went in off the underside of the bar.
Forest came close to a leveller shortly into the second half, when Halford rose highest to meet Reid's corner but headed well over.
Butland was again called into action in the 55th minute when he stopped a goal-bound header from Darius Henderson.
Birmingham finally extended their lead 10 minutes from the end when Nathan Redmond teed up Burke for the striker to side-foot home his second goal of the match.
Butland produced another fine save moments from time to deny Sharp a response for Forest.
However, Forest did manage to find a late consolation when Sharp was brought down for a penalty before getting himself up to convert the spot kick in the 92nd minute.