Liam Buckley's team had goalkeeper Conor O'Malley to thank for saving two penalties - from Jake Keegan and Andy O'Connell - and he was voted as EA Sports Man of the Match.
In front of a packed crowd of 3,662 spectators, both teams set about trying to claim an early advantage but neither could find a way through as stubborn defending proved to be the theme of the first half.
The first real chance on goal saw St Pat's midfielder Greg Bolger test the alertness of Galway goalkeeper Conor Winn on 20 minutes, but the big shot-stopper would save his best for later on.
St Pat's continued to push for the opening goal with Conan Byrne and Sean Hoare both rattling the side netting, while substitute Christy Fagan went close to scoring with his first contribution of the game.
Galway had some notable efforts of their own with Enda Curran forcing O'Malley into a fine save and Jason Molloy shooting just off target following a good period of play.
James Chambers thought that he had finally broken the deadlock with a stunning strike from long range but Winn was equal to it with a truly outstanding diving save.
It ended scoreless at full-time so the game went to extra-time and both teams had great chances to snatch it with Galway's Kilian Cantwell failing to head in and Hoare shooting wide.
A penalty shoot-out was required to decide the winner and O'Malley proved to be the hero as he saved from Keegan and O'Connell while Ian Morris netted St Pat's winning spot-kick to claim a 4-3 shoot-out victory.
Galway United: Winn; Horgan, Oji (O'Connell 103), Cantwell, Ludden; Shanahan, Sinnott, Connolly, Molloy (Walsh 72); Curran (Melody 112), Keegan.
St Patrick's Athletic: O'Malley; O'Brien, Hoare, Browne, Desmond; Bolger, Chambers; Byrne, Brennan (McGrath 90), Greene (Morris 112); Langley (Fagan 66).
Referee: Paul McLaughlin.
