A penalty in first-half injury time gave the La Liga side the advantage, but Barry Ferguson’s men battled to equalise, the best chance coming through Nico Raskin who hit the post.
However, it was the home side who managed to find that crucial second goal inside the final 10 minutes through Nico Williams.
From last Thursday’s first-leg, Ferguson named three changes to his starting XI as John Souttar and Mohamed Diomande returned from suspension while Jefte also came in from the start.
Robin Propper was suspended following his red card last week, while Dujon Sterling missed out through injury. Bailey Rice, who had to come off injured in last week’s first-leg was named on the bench this evening.
Rangers started with Liam Kelly in goal behind a back line of Ridvan, James Tavernier, Leon Balogun, John Souttar and Jefte. Diomande returned to the middle of the pitch alongside Nico Raskin, while Ianis Hagi, Vaclav Cerny and Cyriel Dessers completed the line-up.
For Athletic Club, they made one change from the first-leg, as Alex Berenguer came in from the start, with Inaki Williams on the bench.
With nothing to separate the sides heading into the game, it was an even start to the second-leg inside a sold-out San Mames Stadium. It was the home side who had the first sight of goal as Oihan Sancet tried a strike from inside the box, but Ridvan was positioned well to get a crucial block.
With 12 minutes played, Liam Kelly was called into action to deny Nico Williams with a good save from a low strike, before James Tavernier blocked a follow up effort from Sancet.
Cyriel Dessers had a claim for a penalty shortly after as he appeared to be pulled by Daniel Vivian as he looked to get a shot away inside the box, with the Rangers striker eventually booked for his protests.
There was a blow for the Light Blues with 20 minutes played as Ferguson was forced into a first change of the match with Ridvan unable to continue with Connor Barron introduced in his place.
A cross from the right from Oscar De Marcos almost carved another opportunity for the home side, but Maroan Sannadi sent it wide at the near post.
Rangers were defending well but there was another big chance for the home side in first half injury time as the ball fell for Oihan Sancet inside the box. He took a touch on his chest but thankfully he was unable to keep his effort down as it drifted over the bar.
With just one minute left of first-half injury time, the home side were awarded a penalty for a foul of Maroan Sannadi by John Souttar. It was Sancet who stepped up to the spot and sent Liam Kelly the wrong way with a right-footed penalty low into the bottom corner.
There was a change for Rangers at the break as Clinton Nsiala came on to replace Ianis Hagi.
Just seconds after the break, there was another stop in play as John Souttar and Leon Balogun collided while defending a dangerous cross. The injury forced Rangers into another change with Hamza Igamane coming on to replace the defender.
The Moroccan striker joined Cyriel Dessers up front, with what appeared to be a change of formation for Ferguson’s men.
A set-piece gave Rangers a bit of relief from the home side's pressure. The initial free-kick was blocked but was sent back into the box by James Tavernier. John Souttar headed it back across goal and it fell for Nico Raskin who came agonisingly close as his strike bounced back off the post.
The Gers had a bit of a lift after that opportunity, and Cerny almost found Dessers at the back post with a nice delivery, but it was turned behind by an Athletic Club defender, before minutes later Cerny tried a shot of his own which was gathered at the second attempt by Julen Agirrezabala.
With 10 minutes remaining, Athletic Club had the second goal, and it was Nico Williams who came up with the header at the back post to seal their place in the semi-finals of the competition.
RANGERS: Kelly, Ridvan (Barron, 24), Tavernier, Balogun (Igamane, 49), Souttar, Jefte, Raskin, Diomande (Bajrami, 84), Hagi (Nsiala, 46), Cerny (Danilo, 84), Dessers.
Subs: Butland, McGuire, McCausland, Rice, Curtis, Hutton, McCallion
Athletic Club: Agirrezabala, Vivian, Alvarez, Berenguer (Vesga, 86), Sancet (Gomez, 82), N. Williams (Djalo, 86), Lekue, Ruiz De Galarreta (Prados, 73), de Marcos, Sannadi (Guruzeta, 73), Jaureguizar
Subs: Simon, Gorosabel, Paredes, I. Williams, Nunez, Canales, Boiro.