Los Che had a 2-0 lead for the majority of the game thanks to goals from Santi Mina and Aderlan Santos, but a crucial away goal was scored by Artiz Aduriz with just over 10 minutes remaining, and that was enough to send Athletic through to the final eight 2-2 on aggregate.
The first real chance of the game came inside the opening 10 minutes when Aduriz weaved his way through Valencia's defence before aiming his shot, but Mathew Ryan was able to comfortably save.
After some disappointing results of late, Valencia were able to celebrate on the 13-minute mark when Jose Gaya charged down the left-hand side before delivering a cross into the box.
Los Che did not create many goal-worthy opportunities for the next 10 minutes as they dropped deep to soak up some pressure from their opponents.
Both teams did well to close opponents down off the ball, which made it difficult for either side to break through into the final third.
However, it was not long before Valencia ramped up the pressure and they were unlucky when Andre Gomes's header in the box was deflected off his own teammate Negredo and behind for a goal kick.
In the 37th minute, the home side's pressure counted when Gaya delivered another perfectly-taken cross into the box and found Santos, who sent the ball into the top right corner of the net.
Gomes had a decent opportunity to claim a third goal when Herrerin fluffed his launched clearance, which allowed the Valencia midfielder to hit the ball on the volley, but his effort just went wide.
At the other end, Athletic came close to clawing their way back into the game before the half-time whistle when Aymeric Laporte took advantage of the defenders switching off by thumping a volley towards the goal, but Ryan pulled off a save.
Valencia began the second half as they finished the first - on the front foot. Gaya pulled the strings for the majority of the game and he impressed again by beating a line of defenders and cutting inside before sending the ball to Mina at the far post, but the keeper managed to get there first.
The home side lived dangerously at times as at 2-0 they knew that just one goal from Athletic would send them out of the competition, and that almost happened when Oscar de Marcos unleashed a half-volley from the edge of the area, but he ended up blasting it over the bar.
A glorious chance for Valencia was wasted in the 62nd minute as they struggled to take advantage of Herrerin coming off his line.
Rodrigo managed to get round the keeper before cutting the ball back to Gaya, but the midfielder lifted his shot over the crossbar as he was leaning back when he struck.
The worrying moment for Valencia came in the 76th minute when Raul Garcia flicked the ball on to Aduriz, who tapped it home to give the visiting side that all-important away goal.
Tempers flared in the dugout shortly after the goal and Neville was sent to the stands for arguing with the referee.
Valencia had some opportunities to get back on top and finish off the game but it was not to be.