Fernando Llorente and Oscar de Marcos scored in each half as Athletic overwhelmed United, which netted a late consolation goal through Wayne Rooney, for a 5-3 aggregate victory that should have been much more. The young Basque club, which also mesmerized in a 3-2 win at Old Trafford, reached the last eight of a UEFA competition for the first time in 35 years to emerge as one of the competition favourites.
Valencia drew 1-1 at PSV Eindhoven to advance alongside fellow Spanish club Atletico Madrid, which capped a 6-1 series win with a 3-0 rout at Besiktas.
Schalke, AZ Alkmaar, Hannover and Metalist Kharkiv also advanced, with Friday's draw determining the next round of matchups.
City crashed out on away goals despite rallying at home against its Portuguese opponent, which took a 33rd-minute lead when Matias Fernandez curled home a free kick. Ricky van Wolfswinkel doubled the visitors' lead seven minutes later to consolidate Sporting's 1-0 win in the first leg.
Needing four goals to progress, City almost completed an astonishing comeback as Sergio Aguero scored in the 60th and 82nd, sandwiching Mario Balotelli's 75th-minute penalty, to level the aggregate score at 3-3.
Chelsea is the only remaining English club playing in Europe after reaching the Champions League quarterfinals, with the pair of Manchester rivals having to settle with fighting for the Premier League title, with United leading City by one point.
"I don't think we can complain of the result," United manager Alex Ferguson said. "Trying to analyze this season (in Europe) against other seasons is a bit difficult. It's been a disappointing year."
Ferguson's team failed to respond to counterpart Marcelo Bielsa's tactics, and against another Spanish opponent that thrives off possession and quick touches as Barcelona did in beating the Red Devils in two of the past three Champions League finals.
Lanky striker Llorente powered home a 23rd-minute opener as he spectacularly volleyed Fernando Amorebieta's long ball past United goalkeeper David de Gea, who was left helpless as De Marcos was rewarded for his game-long efforts in leading Bilbao's attack when his shot took a friendly deflection in the 65th.
While Rooney secured some dignity for the visitors with an impressive strike, United was lucky to escape complete humiliation only because of Bilbao's lack of finishing in a second half where it could have added a handful of goals.
"Their work rate is the highest in Europe, their work rate is getting them where they are at the moment, they are showing some good combination football," Ferguson said. "I think they can win it."
While Spanish clubs owe their government a combined €752 million in unpaid back taxes, five remain alive in Europe with Barcelona and Real Madrid favourites in the Champions League.
Valencia defender Adil Rami flicked on Sofiane Feghouli's 47th-minute drive to put Valencia in control in PSV coach Phillip Cocu's debut, with the hosts equalizing in the 63rd through Tim Matavz, who finally beat standout Valencia 'keeper Diego Alves. Rami will miss the next match after being shown a straight red card with five minutes remaining as Valencia followed up a 4-2 home win to take the series 5-3.
Atletico's revival under coach Diego Simeone continued as Adrian Lopez, Radamel Falcao and Eduardo Salvio led the Spanish club past Besiktas.
A late Olympiakos collapse allowed Ukrainian club Metalist to advance as Cristiano Villagra and Marko Devic scored in the final nine minutes to erase Ivan Marcano's opener and see the visitors take the series on away goals after finishing tied 2-2.
Nick Viergever was red-carded after just three minutes and Antonio di Natale stepped up to score the first of two goals in Udinese's 2-1 victory over AZ, which advanced 3-2 on aggregate thanks to Erik Falkenburg's 31st-minute strike.
Hannover reached the quarterfinals of a European competition for the first time in their history with a 4-0 victory against 10-man Standard Liege after the pair drew the first leg 2-2.
Schalke of Germany rallied from a 1-0 loss for a 4-1 victory over FC Twente, as Klaas-Jan Huntelaar's second consecutive Europa league hat trick and Jermaine Jones cancelled out Willem Janssen's 14th-minute opener for the Dutch side.
Paul Logothetis can be reached at: www.twitter.com/PaulLogoAP