Kenny Shiels' Derry made the journey down to Turner's Cross and had to fight until the 95th minute when Dean Jarvis' goal ensured their place in the final four of the competition with a 2-1 win.
Fighting is exactly what The Candystripes had to do in their quarter-final as they went behind inside the opening 10 minutes when Mark O'Sullivan fired in for Cork.
A goal on 38 minutes from Barry McNamee settled Derry's nerves somewhat, but they were made to scrap all the way into extra-time for the victory as a youthful Cork side kept up with them.
In Tolka Park, Craig Hyland was the hero for Shamrock Rovers as he saved three penalties in a shoot-out with Shelbourne following a 1-1 draw that went into extra-time and then to spot kicks.
Rovers, who had Rob Cornwall sent off, had to rely on Dean Clarke to net on 62 minutes to keep them in the tie after conceding an own goal from Danny North just before the half hour.
Pat Fenlon's team held their nerve in the shoot-out with Clarke, North, and Patrick Cregg all proving to be successful, but Hyland took all the plaudits for his superb saves.
It was late drama at the Carlisle Grounds as St Patrick's Athletic claimed a 3-2 triumph over Bray Wanderers with Mark Timlin providing the all-important winning goal.
The Cup holders went ahead through Billy Dennehy on 53 minutes but Bray responded within three minutes as Dylan Connolly found the back of the net. Next to strike was Jamie McGrath for the visitors.
But Gareth McDonagh levelled again for Bray, only for Timlin to send Liam Buckley's side into the semi-finals and a step closer to retaining the trophy that they won last year.