Atletico Madrid also snatched a late win in a Spain-Germany clash, beating Hannover 96 2-1 with a delightful late goal by Eduardo Salvio.
Dutch league leaders AZ Alkmaar beat Valencia 2-1 and Sporting Lisbon beat Metalist Kharkiv by the same score in the other quarter-final ties.
Raul and Fernando Llorente both scored twice to leave Schalke and Athletic level at 2-2 before Oscar de Marcos and Iker Muniain netted in the last 10 minutes for Marcelo Bielsa's team.
Llorente scrambled Athletic ahead in the 20th minute after a mistake by Schalke goalkeeper Timo Hildebrand, only for former Real Madrid striker Raul to prod in Atsuto Uchida's cross two minutes later for the hosts.
Instead, Llorente began Athletic's late rally when he headed in a corner at the near post in the 73rd minute to put La Liga's eleventh-placed team back on level terms.
Schalke suffered more goalkeeping woes as Mathias Schober, who replaced the injured Hildebrand at half-time for his first appearance in three years, failed to hold Markel Susaeta's cross, allowing De Marcos to score from close range.
In a niggly game with plenty of heavy challenges, Mame Diouf slid in Lars Stindl's cross from the right to put Hannover back on level terms seven minutes before halftime and the Germans began to look comfortable.
The former Argentina captain, whose team's fierce tackling and off-the-ball challenges are reminiscent of his playing style, also got involved in a finger-pointing exchange with a Hannover player.
Alvarado repeatedly denied Valencia's forwards, his repertoire including two breathtaking saves from Robe
