Chris Hughton's men are now six points behind third-placed West Ham, but more importantly they extended the gap between them and seventh-placed Middlesbrough to three points by winning their game in hand.
Marlon King opened the scoring in the first half, but Birmingham were pegged back in the second by Danny Ings’s 74th minute strike.
However, Jordon Mutch scored his first goal for the Blues just 90 seconds later, and David Murphy netted a third with three minutes left on the clock.
As expected, Burnley were without Martin Paterson, who limped out of Saturday's 5-1 win over Portsmouth with a hamstring injury. The injury came just six weeks after he returned from a similar problem.
Top-scorer Jay Rodriguez is recovering from a hip problem, so Charlie Austin started up front after scoring his first hat-trick for the club as a substitute on Saturday.
Birmingham made just one change from the side that won 3-1 at Doncaster on Friday, with Wade Elliott replacing Andros Townsend in midfield.
There was a good reception from the Turf Moor fans for former players Elliott and Steven Caldwell before the game. Birmingham captain Caldwell led Burnley to the Championship play-off win over Sheffield United in May 2009, while Elliott was the goalscorer against the Blades at Wembley.
Caldwell almost gifted his former side a goal in the first minute of the game with a misplaced header that gave Josh McQuoid an early sight of goal. However, with Birmingham goalkeeper Boaz Myhill way off his line, the forward failed to get an effort on target.
Birmingham took the lead in the 11th minute, after the Burnley defence had switched off when facing a Murphy throw. Nikola Zigic flicked on, allowing King to net his 16th goal of the season with a first-time effort that deflected off goalkeeper Lee Grant and in.
Birmingham have lost 25 points from a winning position this season, the highest in the division, a statistic that would have given Burnley some hope.
But Eddie Howe’s men could not keep hold of the ball in midfield, and consequently were unable to feed Saturday’s hat-trick hero Austin.
The only chance of note for the home side was a bullet header from David Edgar in the 28th minute from a Kieran Trippier corner, which went wide.
Birmingham started the second half again the brighter side, and King should have scored when set up one-on-one down the right side of the area, but defender Michael Duff came back to brilliantly block.
But Burnley slowly started to gain a foothold in the game, with Ings turning and shooting in the 50th minute, and Myhill looked unconvincing when saving the snapshot.
The introduction of Junior Stanislas midway through the half galvanised the home side. His worrying of the Birmingham defence forced a poor clearance from Myhill, which allowed Austin a chance to control the clearance and then shoot towards goal from 35 yards. However, Curtis Davies was alert and able to clear on the line.
On 71 minutes, Stanlislas then crossed for Ings, whose shot hit the bar, then Austin headed the rebound against the woodwork.
An equaliser seemed inevitable, and it finally arrived two minutes later with Stanislas again instrumental.
The 22-year-old played a one-two down the left with Ross Wallace, and the Scot’s cross was perfect for Ings to head home his second goal of the season, and his first goal at Turf Moor.
It had been all Burnley in the second half, but they then switched off and allowed Zigic a chance, then just 90 seconds later Mutch made it 2-1 with a cool finish after King had been allowed space down the right to hook back the ball into the penalty area.
Birmingham made it 3-1 with three minutes to go, Mutch's free-kick was only half-saved by Grant, and David Murphy had time and space to shoot home the rebound for his sixth goal of the season.