James Hayter put the visitors ahead in the 13th minute with a tap-in, while the Clarets hit back - after a penalty miss from Chris Eagles - through a scrappy Jack Cork effort in the second half.
Rovers will have been the happier side though as they held the Clarets to their third draw in as many games and their seventh in the last 10.
Burnley were on the front foot immediately but Wade Elliott failed to take a glorious chance with just 20 seconds on the clock.
It was Jay Rodriguez who threaded through a flat-footed Rovers defence, but having seen his first-time effort blocked expertly by Neil Sullivan, Shelton Martis' last-ditch tackle denied him a second chance at goal.
Referee David Foster then decided not to award the Clarets what looked a surefire penalty after Wayne Thomas seemingly handled a bouncing Danny Fox cross.
But it was the visitors who claimed a 13th-minute lead against the run of play.
James Coppinger got away down the right and fizzed a dangerous-looking ball into the box.
Cork did get back but his attempted clearance found the post, and with goalkeeper Lee Grant still picking himself up off the turf, it fell invitingly for Hayter, in acres of space, to tap home.
After looking totally in charge the Clarets were rattled and David Healy could have doubled Rovers' lead after wriggling his way into the box, but could only fire tamely into Grant's chest.
Rodriguez could not keep his header down from a Tyrone Mears cross as it fell on the roof of the net.
Burnley were then handed a chance to level in the 38th minute when Dean Shiels was adjudged to have handled as Eagles looked to loft the ball back into the danger zone.
The referee showed no hesitation in pointing to the spot on this occasion, but Sullivan went the right way and pawed away Eagles' powerful penalty.
An in-form Sullivan, 40, then repelled Rodriguez after fine work from Mears to cap a miserable first 45 minutes for the Clarets.
Chris Iwelumo came on in place of Martin Paterson as the hosts looked for a speedy riposte at the start of the second half.
Eagles stung the palms of Sullivan after a good knock-down from Iwelumo, while Clarke Carlisle failed to guide an Eagles corner towards goal.
Fox's shot-cum-cross was then just out of reach for the fast-advancing Iwelumo.
They did, however, equalise in the 63rd minute in scrappy fashion.
Eagles' corner was headed against the post by a towering Carlisle and it was Cork who reacted quickest, just beating Michael Duff to the loose ball, to stab home.
Five minutes later John Oster let fly from 25 yards with an angled drive that Grant did well to tip round the post for a corner.
Burnley were now throwing everyone forward in search of a winner and Carlisle had a swing at the ball, but was well wide after Dean Marney had nodded agonisingly across the face of goal.
But in truth neither side looked like finding a winner as the match finished all square.