Cork City enjoyed a lively opening 10 minutes, getting off to an encouraging start, and should have taken the lead as early as the fourth minute following a splendid move down the left. Gearoid Morrissey linked up with Danny Murphy, whose cross was flicked on by Graham Cummins only for Ian Turner to drill a rising shot off the underside of the crossbar.
Wexford Youths, with a new striking duo in Darragh Walshe and Tom Elmes, led to leading scorer Danny Furlong playing central midfield, with Patsy Malone taking up a central defensive role.
The home side’s defensive set up saw Cork City struggle after that bright opening, although Cummins had a 15th-minute header narrowly wide. The home side were battling but were offering little up front with Cork stand-in keeper James McCarthy rarely troubled.
Cork, however, continued to struggle up front, Vincent Escude-Candau hooking wide when he should have scored.
Within six minutes of the restart Cork City were handed a golden opportunity when Referee McKell awarded a penalty after Ian Turner was taken down just inside the right edge of area but Escude-Candau’s kick was excellently save by keeper Holden.
The long-awaited breakthrough came after 72 minutes when sub O’Neill, only on the pitch for four minutes, fired home a superb 20-meter shot past a helpless Holden in the home goal.
Wexford Youths had Dean Broaders red-carded on 93 minutes as they lost their way and Cork deservedly took their place in the draw for the next round.
WEXFORD YOUTHS: Holden, Phelan, Broaders, Kinsella, Kehoe, Malone (Yelverton 80), Elmes, Wall, Furlong, Walshe (Mullen 80), Vickers (Grincell 66).
CORK CITY: McCarthy, Turner (Morrissey 92), Horgan, Spillane, Murphy, Kavanagh, Morrissey, O’Brien, Duggan, Cummins (Sullivan 81), Escude-Candau (O’Neill 68).
Referee: Jim McKell (Clonmel).