There were also wins for Limerick, Galway and Tipperary
Davy Fitzgerald's Wexford stunned Kilkenny at Nowlan Park to claim a memorable 2-18 to 0-19 win over Brian Cody's charges.
Goals from David Dunne and Jack Guiney gave the visitors a surprise win by 2-18 to 0-19.
Kilkenny keeper Eoin Murphy had a busy day as Fitzgerald's men went hunting goals from the off. Lee Chin also added a touch of class throughout to give Wexford a surprise win and they will now face Tipperary in the semi-final.
Kilkenny’s Lester Ryan and Wexford’s Lee Chin. ©INPHO/Ken Sutton
Elsewhere, Galway progressed to the semi-finals after keeping Waterford scoreless for the last 20 minutes of their clash. Despite two goals from Tom Devine to give Waterford the upper hand, Joe Canning proved to be the hero for the Tribesmen despite seeing red late on.
Galway's Adrian Tuohy and Mark O'Brien Waterford. Image: ©INPHO/Mike Shaughnessy
Canning bagged 1-10 as Galway won by 2-22 to 2-19 and set up a clash with Limerick.
Limerick needed a late onslaught to get them over the line against Cork at Pairc Ui Rinn. Shane Dowling led the scoring for the winners with eight points while David Dempsey added 1-1 along with four points from Gearoid Hegarty.
Limerick's Declan Hannon with Seamus Harnedy of Cork. Image: ©INPHO/Conor Wyse
The scores were level with three minutes left on the clock but Limerick pulled away to advance by 1-20 to 1-18.
Tipperary hammered Offaly 4-28 to 3-13 in a high-scoring encounter at O'Connor Park. Seamus Callanan top scored for the Premier County with 2-11 with Paul Flynn and Stephen O'Brien adding 1-1 each.
Shane Dooley grabbed 2-8 for Offaly but it wasn't enough as the All-Ireland champions progressed to the last four.
In the relegation play-off, Clare condemned Dublin to Division 1B next season with a 3-18 to 0-19 win in Cusack Park in Ennis.
Early goals in the second half from John Conlon and Tony Kelly proved decisive as they retained their Division 1A status for next season.
National Hurling League Quarter-Final Results:
Kilkenny 0-19 Wexford 2-18
Cork 1-18 Limerick 1-20
Galway 2-22 Waterford 2-19
Offaly 3-13 Tipperary 4-28
National Hurling League Division 1A Relegation Play-Off Result:
Clare 3-18 Dublin 0-19
National Hurling League Division 3B Final Result:
Longford 4-19 Warwickshire 3-20 (AET)