The Basque outfit were chasing a first European semi-final appearance in 13 years and started well here with a couple of openings for Oihan Sancet in the opening exchanges.
However, the game’s first flashpoint came down the other end when Dani Vivian pulled down Cyriel Dessers in the box. Despite part of the Nigerian’s shirt being ripped, no penalty was awarded.
Barry Ferguson’s men subsequently soaked up a substantial amount of first-half pressure, as Maroan Sannadi fired wide following some delightful link-up play involving Alex Berenguer and Oscar de Marcos.
Berenguer tried his luck himself after being teed up by Sancet, and saw his curling strike go just past the far post.
Sancet should have broken the deadlock in first-half stoppage time when he fired over from close range, but referee Irfan Peljto pointed to the spot after Harry Souttar brought down Sannadi in the box.
Sancet coolly dispatched the resulting effort from 12 yards, sending Liam Kelly the wrong way just seconds before half-time.
The Gers suffered a further blow right at the start of the second half with Leon Balogun forced off injured, having already lost Rıdvan Yilmaz before the break.
Ferguson’s men responded well to the setback though, and were agonisingly close to restoring parity on the night when Nicolas Raskin’s effort cannoned off the post.
The Glaswegians were enjoying their best spell in the match, and De Marcos had to be alert to head behind Vaclav Cerny's searching cross towards Dessers.
Athletic effectively ended any hopes of a Rangers comeback when Nico Williams powered home a brilliant header at the far post from an excellent De Marcos cross in the 80th minute.
Ernesto Valverde’s side saw out the closing stages with relative ease to book their place in the Europa League last four, where they will face either Manchester United or Olympique Lyonnais.
Meanwhile, Rangers’ European adventure comes to an end having now lost their last five two-legged ties against Spanish opposition.
Flashscore Man of the Match: Nico Williams (Athletic Club)