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.
Raul gave Schalke, third in the Bundesliga, a deserved lead with a spectacular volley from the edge of the area just before the hour and was close to a hat-trick when he chipped the ball just wide.
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.
Teenager Muniain fired in the fourth to complete another impressive win for the side led by former Argentina and Chile coach Marcelo Bielsa, which also won both legs against Manchester United.
Colombian striker Radamel Falcao, scorer of a record 17 Europa League goals last season for eventual champions Porto, headed Atletico ahead early in the tie.
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.
But Atletico retook control after Brazilian midfielder Diego come on just after the hour and Argentine forward Salvio curled in the winner from the edge of the area in the 89th minute.
Diego nearly added a third with an acrobatic bicycle-kick in stoppage time but was foiled by a brilliant stop by Ron-Robert Zieler.
Atletico coach Diego Simeone, dressed in his characteristic all-black outfit, was a lively figure on the touchline, celebrating their winner wildly and coming on to the pitch near the end during a shoving match between the players.
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.
Alkmaar were heavily indebted to Australian Brett Holman and Costa Rican goalkeeper Esteban Alvarado for their win over Valencia.
Alvarado repeatedly denied Valencia's forwards, his repertoire including two breathtaking saves from Robe
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