A thrilling encounter in Germany filled with offensive quality saw the visitors depart with a crucial victory, putting one foot into the semi-finals of the competition
Braces from Fernando Llorente and Raul saw the game level, before the Basque duo finished calmly to give the visitors a significant victory in the final 10 minutes.
After back and forth play with plenty of creativity to start the contest, the breakthrough arrived after 20 minutes.
Markel Susaeta received a low ball towards the right corner, beating a diving Christian Fuchs, before sending in a diagonal pass into the box. Timo Hildebrand could not grab the ball to his chest, fumbling the effort just a few yards out, and Llorente managed to knock home awkwardly from close range after several players went after the loose ball.
But the hosts managed to respond in the 22nd minute when Jefferson Farfan controlled the ball in the box while shielding off a defender, before playing Atsuto Uchida towards the bye-line. The young Japanese defender crossed to the front post, where Raul got a faint touch to beat Gorka Iraizoz and level the match.
For the remainder of the first half, each team advanced intelligently with the ball, playing passes forward and wide to open up space. Both had decent openings but could not create a clear cut chance to enter the break with the advantage.
In the second half, Kyriakos Papadopoulos appeared to have earned the hosts the lead from a set piece, heading on a cross from Jose Manuel Jurado from the middle of the box, but the Greek's effort slid just past the right post.
Then moments later, Raul gave the hosts the advantage when he controlled the ball from 30 yards out, looking to play Huntelaar into the box, where Fernando Amorebieta got a touch on it, sending it into the air.
Raul followed up, drilling a volley from outside the box into the right side of the goal with thunderous power, leaving the Bilbao keeper with no chance, earning his 75th and 76th goals in Europe.
Llorente would match his compatriot's effort, however, heading home at the front post from a corner in the 83rd minute to again level the game.
The remaining action saw plenty of space in attack for both teams, with the Spanish making the most of it. 10 minutes from time, De Marcos tapped home a rebound from a few yards out after substitute keeper Mathias Schober could not control a shot from Markel Susaeta.
In the dying minutes, Muniain made it four on the counterattack, with Susaeta flying down the right before playing the young attacker at the middle of the box, where he finished with ease into the centre of the net.
Marcelo Bielsa's side could celebrate a significant triumph, and with the second leg at San Mames next week, Bilbao look all but certain to be in the semi-finals.