In a typically frantic opening, it was Cliftonville who struck first on 11 minutes. Rhyss Campbell fired over a low cross from the left which rolled across the face of goal where the inrushing Quinn was on hand to convert.
Crusaders slowly began to play their way back into their game with Finley Thorndike's shot being gathered at the second attempt by PJ Morrison, while the Scottish keeper was also equal to Harry Jewitt-White's first-time effort.
The home side came close to doubling their advantage just short of 30 minutes when Ollie Samuels' cross from the left was only half-cleared by Robbie Weir, but the Crusaders skipper recovered to block Rory Hale's shot behind for a corner.
Cliftonville threatened again as goalscorer Quinn dug out a cross from the right which Brian Healy met with his head, but Crusaders goalkeeper Shea Callister was able to spring across his goal to grasp the ball.
In the second period, Hale exchanged passes with Quinn before the Cliftonville skipper fired an angled shot narrowly wide, while Healy had a headed effort ruled off for offside from a Hale cross.
The home side eventually doubled their tally on 63 minutes after Harry Wilson was brought down just outside the penalty area and Dean McMaster curled the resultant fre- kick into the bottom corner of the net.
Crusaders pulled a goal back in stoppage time when Dunlop latched onto a knockdown to fire a shot high into the roof of the net but there was no time for the Shore Road side to force an equaliser.
Crusaders manager Declan Caddell said there were "some positives" despite the opening-day defeat.
"Some of our attacking play was really good, but we switched off twice. The goals we conceded were poor and it cost us," he said.
"We threw everything at it in the last 10 or 15 minutes in the game. It's the first game, there's things to work on and we'll look forward to a home game [against Ballymena United on Saturday]."
