Chris Forrester scored the winning penalty as St Pat's secured their place in the last four after a 1-1 draw at Turner's Cross.
St Pat's keeper Brendan Clarke saved City's opening effort from Garry Buckley and they pressed home that advantage, scoring all five spot-kicks to progress.
Both sides had chances in an open first half but it was scoreless at the break.
Clarke was forced to tip over a fine effort from John O'Flynn in the ninth minute while Ian Bermingham flashed a shot just across the Cork City goal.
The game came to life after the interval.
Clarke denied Danny Morrissey when he was through one-one-one before the visitors took the lead, Jack Bayly heading home after Cork goalkeeper Alan Smith misjudged a cross.
City struck back, however, Liam Kearney finishing the rebound after Rob Lehane hit the post, and went close to forcing a winner from a succession of corners.
Pat's held on for extra-time, though, and when neither side found a goal in the additional 30 minutes, it was up to Clarke and Forrester to break home hearts.