Glentoran came out of the traps the shaper and more hungry side, although they had to wait until the final seconds of the first half to make their dominance count.
The hosts were first to threaten on four minutes as Dylan Connolly squared to Jay Donnelly and his strike was kept out by David Walsh.
Linfield's only meaningful attack of the half almost resulted in the opener with Cooper's low shot palmed on the post by an outstretched Gyollai.
Glentoran pushed for the breakthrough but the Blues defence held firm and restricted them to half-chances.
James Singleton went close when his 25-yard effort took a deflection and flew inches wide of the Linfield woodwork.
Persistence paid off for Declan Devine's men as they finally broke the deadlock seconds into added time.
Linfield failed to deal with Daniel Amos' inswinging corner and the ball fell for an unmarked Kane to blast into the net from three yards out.