The Basque outfit, seventh in La Liga, outplayed Premier League leaders United in the last 16 to triumph 5-3 on aggregate and will next play the fourth-placed Bundesliga side.
Schalke beat FC Twente 4-1 in their home second leg on Thursday after losing 1-0 in the Netherlands.
There will be another Spain v Germany battle after Atletico Madrid, one place below Athletic in the Liga table, were drawn against Hannover, who stand seventh in the Bundesliga.
Atleti were comfortable winners over Besiktas in the last round while Hannover thumped Standard Liege this week to ease through.
Portuguese side Sporting, who clinched their place in the last eight after a nail-biting second leg at Manchester City's Etihad Stadium on Thursday, play Metalist Kharkiv, who overcame a first leg home defeat to Olympiacos to sneak through on away goals.
Sporting are fourth in the Superliga while Metalist stand third in the Ukrainian top flight.
Eredivisie leaders AZ Alkmaar will face Valencia, third in the La Liga but 16 and 26 points off Champions League hopefuls Barcelona and Real Madrid respectively.
AZ triumphed over Udinese in the last 16 while Valencia turned over PSV.
Should Valencia progress, they could face Atletico in the semi-finals after the draw for the last four was also made.
The first legs of the quarter-final ties will be played on Thursday March 29, with the return legs on Thursday April 5.
The semi-finals are also scheduled for Thursdays, with the first legs on April 19 and second legs on April 26.
The final will be played in Bucharest on May 9.
AZ v Valencia
Schalke v Athletic Bilbao
Sporting v Metalist Kharkiv
Atletico Madrid v Hannover
Atletico/Hannover v AZ/Valencia
Sporting/Metalist v Schalke/Athletic Bilbao