The second half was played at the same high-energy tempo as the first, with Glentoran guilty of missing two guilt-edged chances soon after the restart.
David Fisher capitalised on a mistake from Jack Keaney, but bearing down on goal, his tame effort was straight at Ridd.
Ridd then produced a point-blank save with his feet to thwart Jenkins' sweeping effort.
After a lull in the game, Cliftonville went close as Luke Conlan was inches away from finding the top corner on 64 minutes, but his outside of the boot effort flew just over the crossbar.
Gyollai produced a magnificent save to keep Glentoran in the game with five minutes to go, tipping Gormley's header over the crossbar.
Gormley, who netted twice the last time the Reds won this trophy against Coleraine, was not to be denied in extra-time.
He nipped in at the front post to flick the ball beyond Gyollai and into the far corner after a driving run from substitute Corrigan.
Jonathan Russell had a superb chance to take the game to penalties, but he failed to get a good connection on his shot with the goal at his mercy as the ball bounced over the crossbar as Cliftonville held on for victory.