Rovers enjoyed a 1-0 lead at the break thanks to Jordan Rhodes' clinical strike just before the half-hour mark.
Rhodes could and should have doubled their lead ten minutes into the second half as he was sent clear by David Dunn and beat Tom Heaton, but he saw the ball rebound off the post.
Rhodes and Blackburn were left to rue that miss as Burnley suddenly hit the home side with two goals in six minutes.
The first came via captain Jason Shackell as he headed home from close range after Ross Wallace's brilliant free-kick into the six-yard box.
Moments later the winner arrived through star striker Danny Ings. Blackburn failed to clear their lines, and Ings picked up the ball from Ashley Barnes before firing under Paul Robinson - who will be disappointed to have seen the ball hit the back of the net.
The win sees Burnley go eight points clear in second place, following third place Derby's loss to Millwall on Saturday.
Rovers boss Gary Bowyer had made five changes from last week's thumping at Bolton.
Burnley manager Sean Dyche named an unchanged side for the eighth consecutive game, but he had goalkeeper Tom Heaton to thank for keeping the scores level early on, as he got down to scramble away a glancing Rhodes header.
Burnley had the ball in the net after six minutes when Sam Vokes turned in Scott Arfield's cross from close range but offside was given - a close decision.
The Clarets went close again but Kieran Trippier's cross was headed onto the roof of the net by Arfield.
Rovers had lost one of their previous 10 home games in all competitions but could have gone behind after 19 minutes when Tommy Spurr dallied and saw his clearance charged into the path of Ings, who could only fire wide.
But it was Rovers who struck first after 24 minutes. Jason Lowe crossed for Rhodes who did well to make space before clinically hammering into the bottom left corner from inside the area.
Rovers were given a massive let-off six minutes before the break after Craig Conway's backpass found Ings clean through but Paul Robinson saved with his legs.
And they would have been further behind two minutes before half time but Ben Mee got back to clear Lowe's effort off the line.
In a frantic start to the second half, Bowyer's side missed a superb chance when Arfield's poor pass was intercepted and Rhodes was sent clean through but his effort cannoned back off the post and was cleared.
But Burnley turned the game on its head in the space of six minutes.
First, Wallace's superb free-kick was met by a towering header from Shackell after 73 minutes for his second of the season.
And the turnaround was complete when Wallace's low cross was kept alive and the prolific Ings reacted to slide in his 25th goal of the season.
Blackburn rallied and Lowe's injury-time thunderbolt forced Heaton into a fine fingertip save as Burnley held on for their first win against Rovers since 1979.