Udinese progressed with a 3-0 win at PAOK Salonika and PSV Eindhoven demolished 10-man Trabzonspor 4-1 to join them.
Bilbao, under eccentric former Chile and Argentina coach Marcelo Bielsa, struggled to claw back a 2-1 first-leg deficit against the Russians and their cause appeared to be lost when Fernando Amorebieta was given a second yellow card for a trip of Roman Shishkin on the hour.
But only two minutes later Iker Muniain scored at the far post to send Athletic through on away goals.
Former European champions Steaua took a 1-0 deficit to Twente Enschede and their slim hopes ended just before the half-hour following a blunder by goalkeeper Ciprian Tatarusanu, who lost his footing as he tried to control a back pass.
Luuk de Jong pounced before squaring to Nacer Chadli, who calmly rolled the ball into the unguarded net to end the Romanians' hopes.
Udinese, held 0-0 at home by PAOK last week, raced to a 2-0 lead after only 15 minutes thanks to Danilo, who took advantage of sloppy defending to head in, and a deflected shot by Antonio Floro Flores.
Maurizio Domizzi completed the scoring with a second-half penalty.
Tim Matavz scored twice for former UEFA Cup winners PSV against Trabzonspor, who had Tolga Zengin sent off before half-time for deliberate handball. The Dutch side won 6-2 on aggregate.
Hannover 96 won 1-0 at Bruges to complete a 3-1 aggregate win while Valencia's 1-0 win over Stoke City, thanks to a first-half goal from Jonas, sent them through 2-0.
Standard Liege also went through despite a goalless draw at home to Wisla Krakow who had Gervasio Nunez sent off for a second bookable offence just after the hour. The Belgians won on away goals.
