Europa League top scorer Aritz Aduriz got the crucial strike as Athletic Bilbao edged into the last eight on away goals at the expense of Valencia, despite a 2-1 defeat at the Mestalla.
Having won last week's first leg 1-0 at San Mames, Athletic found themselves 2-0 down and staring elimination in the face with 14 minutes to go on Thursday.
But Aduriz's eighth goal of this year's competition made the difference, with Valencia head coach Gary Neville subsequently sent to the stands for furiously protesting against a handball in the build-up.
Athletic came into Thursday's meeting at Mestalla having won 3-0 in the league on the same ground at the end of February, but Valencia were vastly improved and found themselves two up at the break.
Santi Mina opened the scoring in the 13th minute when he was quickest to react to a rebound inside the area from Iago Herrerin's save, before Aderlan Santos' towering header put the home side in front on aggregate.
Valencia appeared to be closing out the victory to seal their place in Friday's quarter-final draw in Nyon, until Raul Garcia's superb flick set up Aduriz, who made no mistake in slotting home to see Athletic through.
Aduriz had threatened to add to his tally as early as the eighth minute, but he could only shoot straight at Mathew Ryan.
Valencia broke the deadlock and levelled the tie five minutes later as Alvaro Negredo's powerful shot was parried by Herrerin into the path of Mina, who converted from close range.
Athletic's hopes were dealt a further blow midway through the first half when influential playmaker and captain Iker Muniain had to come off with what appeared to be a groin injury and was replaced by Markel Susaeta.
The visitors' woe was then deepened as Santos rose to meet Jose Gaya's cross and power a superb header into the top-right corner to give Valencia an aggregate lead.
Andre Gomes whistled a volley agonisingly wide as Valencia went in search of a third before the break, although Ryan had to be alert to deny efforts from Susaeta and Aymeric Laporte.
Following a quiet start to the second period, Gaya lifted his shot over the crossbar when well placed, and that miss proved costly as Aduriz made his telling contribution.
Play was allowed to continue despite Susaeta appearing to handle the ball and Garcia - whose first-half booking rules him out of the next match - improvised superbly to flick the ball over the head of his marker into the path of Aduriz, who calmly converted.
Neville was dismissed for his protestations, and he could only look on as Herrerin made a stunning late save from substitute Paco Alcacer and Dani Parejo bent a stoppage-time free-kick wide to ensure Athletic's place in the quarter-finals.