The woodwork denied Queens Park the opening goal midway through the first half. Rovers failed to close down Bryan Felvus on the edge of the area, and the young forward beat Chris Fahey with a curling right-foot effort only to see his shot rebound off the crossbar.
Rovers took the lead against the run of play when James Mercer rose in the box to head home a pin-point Jered Stirling free-kick.
Then, three minutes later, the visitors doubled their lead. Queens Park keeper David Crawford fumbled a corner and John Bradford hooked the ball home from six yards.
Rovers' third goal came just two minutes later. Crawford fumbled another corner and Mercer was on hand to knock the ball over the line from close range.
Queens Park were denied by the woodwork for the second time in the opening minutes of the second half. Paul Harvey struck a 25-yard effort that was well tipped on to the bar by Rovers' keeper Fahey.
Fahey then pulled off another fine save to deny Spiders' substitute Tony Livingston as time ran out on a miserable afternoon for the hosts.