Bilbao were beaten 2-0 at home by Celta Vigo on Wednesday but the Basque club still made it through to the next round courtesy of the away-goals rule after winning the opening leg 4-2.
Ernesto Valverde's Bilbao took four priceless away goals into the last 16 return fixture at San Mames and they proved invaluable as Celta took control in the second half.
Celta went about reducing the deficit in the 48th minute, with Bilbao defender Xabier Etxeita netting an unfortunate own goal.
Etxeita attempted to clear a Santi Mina cross from the wing but his effort looped over goalkeeper Iago Herrerin.
Celta levelled the tie on aggregate via the penalty spot in the 61st minute.
Referee Juan Martinez awarded a penalty after Mikel Rico had fouled Brazilian striker Charles and Chilean winger Fabian Orellana made no mistake with his spot-kick.
But Celta could not complete a sensational comeback as Bilbao survived to qualify for the quarters, where they will meet Malaga.
Sevilla secured a showdown with Espanyol after easing past Granada 6-1 on aggregate.
Frenchman Kevin Gameiro bagged a double to inspire the home side to a resounding 4-0 victory at the Estadio Ramon Sanchez.
Gameiro broke the deadlock in the 18th minute, side-footing past Granada keeper Roberto.
Iago Aspas got in on the action nine minutes later when he looped a header over Roberto before Gameiro and Denis Suarez completed the scoring in the second half.
Meanwhile, Villarreal and Getafe also won through to the quarter-finals.
A 2-2 draw at Real Sociedad was enough to see 10-man Villarreal triumph 3-2 on aggregate.
Next up for Villarreal are Getafe, who were 2-1 winners over two legs after accounting for Almeria 1-0 on Wednesday.
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