The home side went close to breaking the deadlock midway through the first half when David Weatherston took the ball round the keeper before seeing his weak effort cleared off the line by Graham Hay.
But Queens didn't have to wait long before opening the scoring, as Paul Harvey netted with a thunderous 20-yard effort into the top left corner after 26 minutes.
The goal gave Queens some added confidence and Mark Ferry saw a 20-yard effort parried by the goalkeeper before heading wide minutes later.
The hosts put Montrose under serious pressure at the start of the second half and their superiority paid dividends when Brian Felvus netted with a low finish from 10 yards.
But the visitors pulled a goal back within 90 seconds with their first shot on target, as the Queens defence went to sleep.
Gary Middleton swung a deep free-kick to the back post where the unmarked Greig Henslee netted with a neat side-footed volley.
Montrose were unlucky not to level midway through the second half when Henslee saw an effort well blocked by Mark Cairns, before substitute Kris Brash struck the woodwork.
But the visitors did eventually equalise in the 79th minute when Stephen Fraser fired home from James Russell's cutback.