A badly depleted home side made light of five enforced changes to take the lead after just 10 minutes.
Cliftonville failed to deal with a high ball into the box, allowing Aidan Steele to crash a volley off the crossbar from the edge of the penalty area.
Hawthorne was quickest to react to the rebound, nipping in to steer the ball low under body of helpless Reds' keeper PJ Morrison from six yards.
The lead lasted barely five minutes as the visitors struck for the equaliser after a quarter of an hour.
Quinn's inswinging corner eluded everyone except Cliftonville defender McClelland, lurking at the back post, who had the simple task of heading into an unguarded net.
Cliftonville went in front for the first time just after the half hour thanks to a fine individual effort from the impressive Quinn.
Liam McStravick showed great awareness to spot the run of his team-mate, who waltzed his way past a number of defenders before slipping the ball past Gartside for his fourth goal in six games.
Carrick started the second half on the front foot and came within the width of a post of levelling as Steele's looping effort from a Kym Nelson cross rebounded off the woodwork with the goalkeeper beaten.
They did have the ball in the net just after the hour when Danny Gibson poked home after Joe Crowe's initial effort had been saved by Morrison, but the effort was ruled out for offside by a linesman.
All their efforts were in vain as Cliftonville struck for a decisive third goal in the dying minutes, McMaster curling home a beautifully struck free-kick from 25 yards, to make it back-to-back wins for Jim Magilton's side.