France centre-back Adil Rami scored twice in two first-half minutes to put Valencia 2-0 up against AZ.
Valencia went on to add two more goals through Jordi Alba and Pablo as they overturned their one-goal first-leg deficit with ease.
Valencia players celebrate after confirming their passage to the Europa League semi-finals.
Schalke captain Raul upheld the tradition of every new club to play at the San Mames leaving a bunch of flowers by Athletic's statue of Pichichi.
Klaas-Jan Huntelaar netted his 14th goal in 12 Europa League games this season to give Schalke the lead against Athletic.
Ibai Gomez scored a superb goal for Athletic as they pegged Schalke back to 1-1. The Spaniards were defending a 4-2 first-leg lead.
Veteran striker Raul was on target to restore Schalke's lead with a brilliantly-taken strike.
But Athletic hit back again through Markel Susaeta to draw 2-2 and advance with a 6-4 aggregate victory.
Atletico Madrid won their home leg against Hannover 2-1 and took the lead in Germany thanks to Adrian's goal.
Former Manchester United striker Mame Biram Diouf kept Hannover's dream alive with a late leveller.
However, Falcao grabbed his customary Europa League goal to secure Atletico a semi-final showdown with Valencia.
Ricky Van Wolfswinkel leapt highest to head Sporting in front against Metalist and put them 3-1 up on aggregate.
Jonathan Cristaldo equalised for Metalist but Cleiton Xavier missed the chance to take the game to extra-time from the spot.