Wexford 1-16 Galway 0-14, Nowlan Park
A 63rd-minute goal from Eoin Quigley was the turning point at Nowlan Park as Wexford beat Galway 1-16 to 0-14 to qualify for the NHL Division One semi-finals.
John Meyler's side set up a Thurles clash with Kilkenny next Sunday after coming good in the second half.
The sides were level eight times in all, and locked at 0-9 apiece at half-time.
Two superb saves from Wexford captain Damien Fitzhenry were crucial, while Stephen Nolan (0-3) shone as a target man for the Models.
Iarlaith Tannion finished with 0-4 for Galway, but their scores dried up in the second half allowing Wexford to shoot themselves into the last-four.
Ger Loughnane's Galway side made the better start with opening points from Tannion and Eugene Cloonan, who top-scored with 0-6, inside a minute.
Damien Hayes made it 0-3 to 0-0 in Galway's favour in the fifth-minute, but Wexford soon settled with three successive scores from Paul Carley, Nolan and Michael Jacob.
Cloonan (free) and Alan Kerins edged Galway back in front but two superb points from Diarmuid Lyng had Meyler's men level again at 0-5 apiece in a lively encounter full of good hurling.
Once again, a Cloonan free nudged Galway ahead and he added another soon after to leave two points between the sides. Nolan immediately reduced the deficit and it took a brilliant save from Wexford keeper Fitzhenry to deny what looked a certain goal for Kevin Broderick.
A 25th-minute free from Carley then squared up the tie for a third time, Cloonan and Michael Jacob, who was neatly set up by Darren Stamp, traded scores before Wexford led for the first time thanks to an over from Rory Jacob.
After wides from Carley and Mark Kerins, Tannion, who was part of Limerick IT's Fitzgibbon Cup-winning side last month, ended the half's scoring, after a lengthy build-up from the Galway forwards, to leave the sides all square.
Loughnane sought to add some bite to his attack by introducing David Tierney and Niall Tierney for Mark Kerins and Broderick for the start of the second half.
Healy did have an immediate impact, pointing Galway into a short-lived lead, four minutes after the restart, but the men from the west could only add four more scores as Wexford began to show better control in front of goal.
Nolan, who received a standing ovation when he was taken off in the closing stages, made it 0-10 apiece before two Tannian scores had Galway leading again, and although Carley was again wayward with some placed balls, the sizeable Wexford support were soon on their feet as their side edged clear.
Barry Lambert got them level at 0-12 apiece before three successive points from the Models, with Mitch Jordan, Stamp, who slapped over having moved tellingly to centre-forward, and Rory Jacob, with a lovely left-handed point, all bisecting the posts.
Then came Quigley's goal as Wexford added some cushion to their lead. Cloonan ended a 20-minute scoreless spell for Galway with a point and Alan Kerins followed suit, but a defiant Fitzhenry was in unbeatable form and Cloonan was denied a late goal when the Duffry Rovers stopper batted away his late free.
A monster point from Willie Doran from his own half completed the scoring for Wexford who now move on to renew rivalries with the Cats, whom they lost to by 1-23 to 2-12 in last summer's Leinster SHC final.