Sharp struck with a superb 79th minute header to pile on the misery for Wednesday who have now gone eight games without a victory.
But they were desperately unlucky not to take anything from the match after outplaying a below par Doncaster side for long periods.
Wednesday were the better side in the first half and had far more of the play overall, but lacked the killer instinct that Sharp provided for Rovers late in the game.
Despite their poor recent record, Wednesday and Laws were given tremendous backing by their fans as they out-battled Rovers in the early stages.
But it was Doncaster who almost snatched the lead against the run of play in the 29th minute when Dean Shiels beat visiting keeper Lee Grant with a curling shot that sailed inches wide.
Darren Potter had a goalbound effort blocked for the Owls before Sean McAllister had an effort saved after home skipper Martin Woods had given the ball away.
Wednesday should have scored just before the interval when Leon Clarke dribbled clean through but fired just wide with only the keeper to beat.
There was little to choose between the sides in the second half until Sharp broke the deadlock after latching onto a right-wing cross from substitute Simon Gillett eleven minutes from time.
Wednesday battled to the finish but Doncaster defended well and their final effort came and went deep into injury time when Mark Beevers volleyed just over the bar.