Nationwide Conference top scorer Paul Barnes notched his 20th goal of the season as Doncaster clinched their first win in three home games.
There was some dispute about the goal with Justin Jackson claiming he had the final touch but the ball looked over the line when he made contact.
It was far from convincing for Doncaster who produced another patchy home display and Burton looked the better side for periods of the first half.
But the visitors were disappointing after the interval and had nothing to come back with once Rovers had scored.
Albion almost snatched the lead straight from the kick-off when Christian Moore headed narrowly wide from Terry Henshaw's cross following a move involving several players.
Craig Dudley had a low shot cleared moments later as Doncaster struggled to settle before Darren Stride missed a great chance when he headed wide.
Burton looked fragile at the back but Doncaster were unable to take advantage and Paul Green wasted a great chance when he blasted wide after Barnes and Jackson had combined in the 33rd minute.
Ricky Ravenhill landed Doncaster's only shot on target in the first half but his 42nd minute volley was comfortably saved.
Rovers finally made the breakthrough in the 54th minute thanks to some ingenious play by Green who dribbled the ball along the byline following a corner before cutting it back for Barnes to fire home.
Barnes had a shot on the turn brilliantly saved by Lee Camp five minutes later before Burton had appeals for a penalty rejected when Moore was sent tumbling after being put clean through by Clough.
It still needed a fine save from Rovers' keeper Andy Warrington in injury time when he denied substitute Sean Farrell before the points were safely in the bag for a jittery Doncaster side.