Marlon King fired his 16th of the season to give Blues an early lead in a first-half of few chances.
The home side were sparked into life by the introduction of Junior Stanislas before the hour and forward duo Danny Ings and Charlie Austin both hit the crossbar before the former levelled.
Chris Hughton’s men responded instantly and Jordon Mutch netted his first goal for the club with 15 minutes to play, leaving full-back David Murphy to apply gloss to the scoreline when Burnley goalkeeper Lee Grant dropped Mutch’s cross.
Birmingham are now three points clear of seventh-placed Middlesbrough, who are outside the top six on goal difference, having won their game in hand on the clutch of teams on 63 points.
At the turn of the year Eddie Howe’s Clarets also harboured play-off ambitions, but they remain 16th and on a run of four wins in 16 matches.
Ross Wallace was utilised as an emergency left-back in the absence of injured duo Ben Mee and Danny Lafferty, while Austin’s hat-trick from the bench in Saturday’s 5-1 victory at Portsmouth earned him a start in attack alongside Josh McQuoid.
In Birmingham’s only change, Wade Elliott was recalled to midfield to face his former club in place of Andros Townsend, who dropped to the bench.
Ings, who started the game in an unfamiliar role at the tip of a midfield diamond, scuffed an early half-volley wide from the edge of the area before the visitors took an 11th minute lead.
Nikola Zigic easily won a flick-on from Murphy’s throw-in and King was allowed ample room in the box to fire goalwards.
Grant got a strong arm to the shot and was unfortunate to have the ball ricochet off his head and into the net.
Burnley defender David Edgar might have brought his side level when he was afforded a free-header from Kieran Trippier’s inswinging corner.
King had the chance to double his tally on the resumption when he charged clear of Edgar’s attentions onto Chris Burke’s through-ball, but some untimely hesitation allowed Michael Duff to make a last-ditch tackle.
Dean Marney hobbled out of the action following a challenge with Zigic and the introduction of Stanislas along with a switch to an attacking 4-4-2 revitalised Burnley.
Former West Ham winger Stanislas was involved as the Clarets came close to a spectacular equaliser in the 66th minute, springing the offside trap to bring Myhill from his area to clear. With the goal unguarded, Austin launched a speculative effort from distant that Blues centre-back Curtis Davies headed behind.
They came closer still and Stanislas was involved once more, storming into the right-hand side of the area for Ings to thump his cutback against the bar and Austin to nod the rebound back onto the woodwork.
The writing was on the wall when the equaliser arrived in the 74th minute. Stanislas popped up on the left this time to tee-up Wallace and his inviting cross was powered home by the unmarked Ings.
Birmingham immediately sprung from their slumber and, after Zigic drilled narrowly wide, they were back in front.
King outmuscled Edgar in typical fashion before racing into the box and picking out Mutch, who applied the finish.
Blues sealed the points when Murphy capitalised on Grant’s 88th blunder from close range, before Ings blazed over when given the opportunity to set up an exciting finish.