The Old Firm giants, already crowned champions of the division, were always in control against a side currently second in the table.
David Templeton broke the deadlock in the 18th minute when he latched on to a loose ball at the edge of the box - although keeper Neil Parry did get a flailing hand to it.
As Rangers continued to dominate against their Glasgow rivals Lee McCulloch fired a free-kick wide of Parry's left post.
And 10 minutes before the break, Rangers doubled their advantage. From Ian Black's lofted free-kick, McCulloch's weak seven-yard shot was mishandled by keeper Parry and Fraser Aird gratefully converted from close range.
Queen's Park, however, did have their moments during the opening hour of the contest.
Jamie Longworth was a threat early on while Ricky Little smashed a 25-yard free-kick inches past a post.
Blair Spittal came mighty close with a 59th-minute effort as the hosts enjoyed a spell of dominance early in the second half.
But Rangers killed off their opponents with a clinical third goal in the 69th minute, as Templeton slipped the ball to the unmarked Kane Hemmings who curled home from the edge of the box.
Five minutes from time Templeton netted his second of the afternoon, finding the bottom right-hand corner of the net from 30 yards.
The hosts finally made the scoresheet shortly afterwards when Tony Quinn's deflected cross was headed home inside the six-yard box by 17-year-old Lawrence Shankland.
However, it was the home side's 10th straight defeat at the hands of Rangers whose most recent loss to Queen's Park came during the 1947/48 season.