Jayden Reid gave Blues an 18th minute lead and they looked to be set fair against the table-toppers.
But Burnley struck in the second-half to consolidate their position and Blues dropped to fourth in the standings, five points from Saturday's visitors.
Blues got off to a decent start, taking the lead after breaking swiftly from deep.
Jack Concannon slipped a lovely pass into the path of Reid just over halfway and he carried towards goal before firing an angled right footer that was too hot for the 'keeper to keep out.
Then, from a left wing throw, Blues worked the ball back nicely to the edge of the area for Caolan Boyd-Munce and he sent in a powerful, curving shot that Lukas Jensen tipped away at full stretch.
After the break Moha Ramos was called into serious action when he had to deal with a difficult dipping shot by Johann Berg Gudmundsson and then recover ground quickly to block the follow-up.
Burnley levelled through Lewis Richardson, who profited when a looping throw into the box bounced off Nico Gordon to him and he cracked a fierce shot past Ramos.
The visitors went ahead through another piece of good fortune when a move down the left flank saw the resulting cross go in off Joe Redmond as he tried to adjust and clear six yards out.
Burnley struck the crossbar after Blues got caught playing out before Burnley's third goal came from a back post corner, Oliver Younger heading in.
Blues: Ramos, Gordon, Forrest, Campbell, Redmond, Roberts (Thompson-Sommers 76), Medina, Boyd-Munce, Reid, Bailey, Concannon (Andrews 76). Subs not used: Bradley-Hurst, Fernandez.