After a quiet opening at Solitude, the Glens came closest to breaking the deadlock after Ryan Cooney let fly from 25 yards, but his dipping, curling effort was pushed to safety by an alert Lewis Ridd.
Declan Devine’s men continued to threaten from set plays and went close as MJ Kamson Kamara met Danny Amos’s inviting corner, but failed to hit the target.
That pressure should have told minutes later when Pat Hoban out-muscled Shaun Leppard and raced through with Lewis Ridd to beat, only for the Cliftonville goalkeeper to stand tall and deny the former Derry City striker.
After the interval, the away side maintained their momentum as Ross Clarke cut in from the right to exchange passes with Aaron McEneff before seeing his right-footed strike from 20 yards tipped behind by Ridd.
The breakthrough arrived just after the hour mark when Pat Hoban converted coolly from the penalty spot after Joe Toole’s high challenge on Aaron McEneff prompted Tim Marshall to point to the spot.
The visitors doubled their advantage midway through the second half as MJ Kamson Kamara diverted Ryan Cooney’s corner into the path of substitute Stewart, who pounced to head home from close range.
The points were effectively secured five minutes later when Stewart collected possession from McEneff under no pressure and curled into the far bottom corner.
Deep into injury time, Hoban rounded off a memorable night for Declan Devine’s men after latching onto a mistake from the Cliftonville rearguard and slotting home his 19th league goal of the season.