Rangers defender Bilel Mohsni appeared to hamper Rovers goalkeeper Neil Parry in an aerial challenge as he attempted to reach a Lee Wallace cross and the ball ended up in the back of the net.
League Two Rovers took the lead on 14 minutes when Lee McCulloch headed the ball to Ciaran Donnelly when attempting to clear a corner and the midfielder cushioned his volley over the line from four yards.
Rangers then spent the best part of an hour peppering the Rovers penalty area with high balls in an attempt to restore parity.
When the League One leaders did vary their play they found a resolute Rovers defence marshalled by their tax inspector goalkeeper Parry.
David Templeton tried to catch Parry out with a curling shot towards his front post but the keeper was right behind the ball.
Ricky Foster then whipped in a cross to the near post which just evaded Nicky Law and had Parry scrambling to clutch the ball at the second attempt.
Dean Shiels was then thwarted by the unsighted Parry as he got down and clawed away a net-bound shot, which was cleared by Ross Dunlop.
McCulloch fired narrowly wide in the opening minute of the second half after the ball broke off Mohsni at a corner, and Black shot straight at Parry's shortly after that.
Jon Daly then missed a sitter when yet another long punt dropped in front of the striker at the back of the box and he pulled his shot well wide of the target.
Daly had a low shot saved by Parry and Templeton's volley clipped the top of the crossbar from 12 yards.
In between those chances Cammy Bell had to punch the ball from under his bar from a Gary Phillips corner just after the hour mark.
The winners of the replay - which has been pencilled in for New Douglas Park on March 18 - will take on Dundee United in the semi-finals.