The victory was no more than Rovers deserved for a gritty display after falling behind to a Charlie Austin goal in the 43rd minute.
Theo Robinson equalised with a curling effort which looked more like an attempted cross in the 48th minute and Rangers were unable to raise their game and restore their lead.
Instead it was Doncaster who stepped up the pace and they were rewarded when Quinn headed his first goal for the club in fine style following a corner.
There was very little threat from either side in the first half as Rangers displayed plenty of patience as they built up their attacks while Doncaster defended comfortably.
Rangers had more possession but Rovers were largely untroubled and Ross Turnbull saw little of the ball in the home goal.
It was not until the 17th minute that either side had a sight at goal and it was Doncaster who provided the first moment of menace.
Winger Mark Duffy dribbled his way inside from the left and hit a shot on target from 20 yards but Robert Green made an easy save.
Niko Kranjcar had a great chance to put Rangers in front in the 26th minute after latching on to a deflected pass from Danny Simpson but he blasted wildly off target from close range.
Doncaster started to enjoy a spell on top just before the interval and Duffy had a shot blocked by Richard Dunne after Federico Macheda had put him through.
But the home side were rocked by a goal against the run of play a couple of minutes later.
Rangers broke quickly out of defence and Jermaine Jenas managed to prod the ball through to Austin whose fairly tame shot appeared to be covered by Turnbull all the way.
But the Rovers keeper inexplicably allowed the ball to slip through his fingers and it trickled into the net to give Austin his ninth goal of the season.
The home side was straight on the attack at the start of the second half as they battled to get back in the match.
But Turnbull's confidence was tested first when Matt Phillips tried his luck with a speculative effort.
This time the Doncaster keeper made a comfortable save.
Like the Rangers goal, Doncaster's equaliser had a touch of good fortune about it when it came in the 48th minute.
Outstanding skipper Ritchie Wellens picked out Robinson with a superb pass and the Doncaster striker curled the ball into the net from a narrow angle on the right when it looked as if he was trying to cross the ball.
Macheda took a fierce shot into the side netting and Bongani Khumalo headed over from a free kick as Rovers pressed for a winner.
It came in the 89th minute when Quinn rose to head home a corner from Duffy and Wellens almost added a third but Green did well to parry his cracking effort in stoppage time.