The result gave Marcelo Bielsa's effervescent young side a richly deserved 5-3 aggregate victory, following last week's equally impressive 3-2 win in the first leg at Old Trafford.
Sir Alex Ferguson's team were bidding to emulate Chelsea's stirring comeback in their Champions League game with Napoli on Wednesday, but they fell to a stunning volley from Fernando Llorente and a second-half strike by Oscar de Marcos.
United replied with a fine goal from Wayne Rooney in the 80th minute, but the visitors could not find the two goals they needed to take the game to extra time.
It was a chastening experience for last season's Champions League finalists, who will now have to plough all their resources into the defence of their Premier League crown, having stolen in front of title rivals Manchester City last weekend.
Bilbao, currently seventh in La Liga, will learn their quarter-final opponents in Friday's draw in Nyon.
United had been bewildered by the pace of Bilbao's attacks and the slickness of their passing in the first leg and the visitors were quickly hustled out of their stride at a raucous San Mames by the home side's aggressive pressing.
Bilbao carved the United defence apart in the 14th minute but when Iker Muniain's skidding shot bounced back off the left-hand post, de Marcos could only hoist the rebound high over the crossbar.
It was a warning that United failed to heed.
Four minutes later, Jon Aurtenetxe sent a 60-yard pass over Rio Ferdinand's head and Llorente met it with a magnificent volley from just inside the 18-yard box that flashed past David de Gea's outstretched right hand.
The goal left United needing to score three times to go through and moments later Ryan Giggs saw a goal-bound header deflected over the bar by a red and white shirt.
Llorente had been United's tormentor-in-chief at Old Trafford but an injury forced him to leave the fray in the 40th minute, with Gaizka Toquero taking his place.
The sight of Bilbao's talisman trudging from the field was a boon for the away side but Bielsa's men continued to attack with brio and Andoni Iraolo almost scored an exquisite solo goal early in the second half.
The right-back picked up the ball on the edge of the box and then tricked his way past Jonny Evans and Rio Ferdinand in turn, only to place the ball the wrong side of the right-hand post from 10 yards.
Toquero fired over shortly afterwards but in the 65th minute de Marcos put the result beyond doubt with a left-footed shot that flicked off Rafael to leave de Gea clutching at thin air.
To United's dismay, Bilbao refused to be satisfied, and after Markel Susaeta had side-footed wide from a Javi Martinez cut-back, Toquero headed onto the roof of the net from Iraola's right-wing cross.
Rooney finally gave the travelling support something to cheer about when he arced a fine strike into the top-right corner from 20 yards, while substitute Danny Welbeck also tested Bilbao goalkeeper Gorka Iraizoz with a low shot from distance.
Source: AFP
Source: AFP