A goal after 18 minutes by Gary Caldwell settled the home team and another by Scot McDonald just before the interval looked to have removed any chance of an upset.
However, Queens Park attacked Celtic from the start of the second half and gained their reward when substitute Adam Coakley reduced the deficit.
The sending-off of Coakley after being booked for his goal celebrations and then a soft foul on Lee Naylor probably spared Celtic an even more embarrassing scoreline.
Queens threatened early on when Stevie Nilcholas' deflected shot had Artur Boruc scrambling across his goal.
In their first attack, Celtic went so close to scoring but Jan Vennegoor of Hesselink's header was superbly cleared off the line by James Brough.
The home side took the lead when Darren O'Dea headed a cross by Shaun Maloney back across goal for Caldwell to send a powerful header into the top corner.
The home side were in command and Maloney and Aiden McGeady both tested David Crawford with powerful shots.
Mark Crosas limped off to be replaced by Massimo Donati after a strong challenge by Paul Harkins before Celtic added a second.
McGeady's cross to the back post found Vennegoor of Hesselink whose floating header was nodded home by McDonald from close range.
At half-time it looked all over for the visitors but from the start of the second half they peppered Celtic's goal with long-range shots.
The pressure eventually paid off when a cross by Ryan Holmes was met at the neat post by Coakley and although Boruc blocked his first effort, the substitute slid home the rebound.
Crawford did well to beat out Vennegoor of Hesselink's powerful shot before McDonald shot wide.
Even though Queens Park were down to ten men, they nearly equalised late on but Paul Cairney's free-kick was deflected over and Richard Sinclair's volley from eight yards was off target.