The Crues were heading for a share of the spoils with the game scoreless with just minutes to go but Danny Morrissey popped up with a late winner at Turner's Cross.
Cork enjoyed the majority of the game with only a Paul Heatley effort in the first half going close for the visitors.
The tie is still evenly balanced and all to play for at Seaview next Monday night in the second leg.
Lisburn Distillery earned a visit to Crusaders on Saturday after defeating a battling Dundela side 2-1 at New Grosvenor in their Irish Cup sixth round replay.
In the previous attempt to play the tie fog forced the game to be abandoned with the Whites leading 3-0 and Dundela made the most of the opportunity to start from scratch again by taking an early lead through Ben Roy.
Aaron Boyd levelled the tie before Alan Davidson made it 2-1 for the Whites to set up a quarter-final tie at Seaview.
Two second half goals from Glentoran gave them a 2-1 victory over Glenavon at The Oval and moved Eddie Patterson's men up to fourth place in the Danske Bank Premiership.
Glenavon had a great chance to take the lead in the first half from the penalty spot, Guy Bates struck the spot kick well but Elliott Morris guessed right to make the save.
The Lurgan Blues went close again when Andy Kilmartin and Brendan Shannon linking up well before the latter brought a good save out of Morris.
The visitors were by far the more creative in the first half and failed to make another chance count when Bates connected and then right on the half time whistle Shannon had an effort saved by the busy Morris.
Glentoran played much better in the second half and two goals in five minutes secured the points.
Andy Waterworth struck his 15th goal of the season with a hard low drive that flew past Davy O'Hare on 67 minutes. Glenavon had hardly time to recover when they were rocked again, this time a superb free kick by Stephen Carson went in off the post.
Glenavon did pull a goal back in added time when Ciaran Martyn steered home a cross from Gary Hamilton.