Sir Alex Ferguson's men again outplayed by the Basque side and were never close to qualifying for the next round as they are thoroughly beaten by Marco Bielsa's side
Athletic took a commanding lead through a wonderful goal from Fernando Llorente and Oscar de Marcos, while United hit back with a wonder goal late on from Wayne Rooney.
The hosts were on the front foot for almost the entire match, as United were put to the sword by some high intensity play with plenty of technique on display.
Sir Alex Ferguson opted to go with Rooney as a lone striker, backed up by Ryan Giggs in an attacking role, Ashley Young was started on the left wing with Park Ji-Sung on the right and a central midfield pairing of Tom Cleverley and Michael Carrick.
Athletic lined up with a similar team to that which impressed at Old Trafford, with Iker Muniain and Llorente the main attacking threat for the hosts.
United attacked straight from the off, as Park found Rooney down the left, but the striker got his chip wrong as it sailed across the face of goal.
It looked certain to be an end-to-end game, with the home side swarming around the ball when United were in possession, and attacking with vigour when on the ball.
Young was the next player to go close, put through by Cleverley but seeing his shot blocked before Marcelo Bielsa's men almost put the tie beyond the Red Devils.
Llorente played a neat reverse pass through to Muniain, on the edge of the area, whose low drive cannoned back off the and back to De Marcos, but the 22-year-old conspired to heave an effort over the bar with David de Gea out of position.
But it wasn't long before the visitors were behind, to a fantastic finish by Llorente. A simple long ball over the top from Fernando Amorebieta exposed Rio Ferdinand, and the Spanish striker volleyed across the face of goal and into the far corner, giving De Gea no chance.
It meant that United went 4-2 down on aggregate and would need at least three goals to advance to the round of 16.
A fabulous cross from Young to the back post five minutes later found an un-marked Giggs, but his header was brilliantly deflected over the bar.
United continued to press, Cleverley controlled a ball on the edge of the box and took a knock on the leg from a Bilbao defender, but nothing was given as the midfielder went down fairly easily.
The half ended with both sides already nursing injuries, as goal scorer Llorente was substituted five minutes before the end, and Rafael fell awkwardly on a soft patch of grass, but seemed to recover before the break.
Going into the second half Ferguson's side still needed three goals, and didn't look like getting close as Athletic bombarded them.
The hosts were inches away from an astounding goal as Andoni Iraola tricked his way into the area, rolling the ball past Carrick then Ferdinand, but placing his shot just wide of the post, with the home crowd about to take the roof off the stadium.
Time and again United were forced to kick the ball long under immense pressure from the hosts, and they could not get their own game going with Rooney looking out of sorts after little time on the ball.
Ferguson perhaps gave a telling sign of how he thought the game was progressing just after the hour mark when he withdrew Ferdinand and Carrick for Chris Smalling and the promising but inexperienced midfielder Paul Pogba.
And two minutes later United's European misery was all but sealed. A cross in towards Gaizka Toquero was well blocked by Smalling, but the rebound fell to De Marcos 10 yards out, and his shot was deflected past De Gea by the outstretched foot of Jonny Evans.
Even as the match progressed and Danny Welbeck was introduced, United continued to struggle and failed to create any chances.
It was, in fact, Athletic who were coming closer and closer to adding a goal, as Iraola's great cross was inexpleicably headed over from six yards by Toquero.
It was another poor European night for United, and the final one of the season for them. Rooney went down with what looked like a nasty tackle on his ankle, but recovered to carry on.
Rooney recovered so well, in fact, that he was able to curl a beautfiul effort into the top corner to bring a goal back for the visitors that was even applauded by the home fans.
United were starting to look like they had a way back into the game, with Rooney creator-in-chief but De Marcos almost snatched another goal with a little under five minutes remaining.
Welbeck came close from long range, forcing the first save of the night from Iraizoz, but the keeper did well to push it around the post.
Ultimately though, United were nowhere near their peak, and suffered as they were outplayed by the hosts in every department, and deservedly knocked out of European competition for a second time this season.