The Spaniards claimed a deserved 3-2 win in the first leg of their last-16 tie, despite falling behind in the 22nd minute to Wayne Rooney's effort.
Fernando Llorente headed Athletic level on the stroke of half-time and an excellent finish from Oscar De Marcos put them in front in the 72nd minute.
Iker Muniain took advantage of some terrible defending from Rafael to add a third in the 90th minute, but Rooney converted a penalty in stoppage time to make United's task in the return leg a little less daunting.
United's Premier League title rivals Manchester City also suffered a shock 1-0 defeat at Sporting Lisbon.
Brazilian defender Xandao ensured the Portuguese side would have a lead to defend in the return match with a clever backheeled finish after 51 minutes, after City goalkeeper Joe Hart had brilliantly blocked his first attempt.
PSV Eindhoven scored twice late on to keep their hopes in the competition alive, despite being totally outplayed by Valencia in their 4-2 defeat at the Mestalla.
Roberto Soldado scored twice, the second a penalty, after Victor Ruiz's opener as the hosts stunned PSV with three first-half goals.
The Dutch side looked down and out when Pablo Piatti added a fourth - the 10th goal PSV have conceded in five days following their 6-2 Eredivisie defeat by FC Twente on Sunday - but an Ola Toivonen penalty in the 83rd minute and another goal from Georgino Wijnaldum in the 90th minute got them back in the tie.
Atletico Madrid raced into a 3-0 lead inside 37 minutes of their first leg at home to Besiktas, but had to settle for a 3-1 win.
Eduardo Salvio scored twice in four minutes to put the hosts in control and they were cruising when Adrian Lopez added a third.
But Simao Sabrosa, playing against his former club, pulled a goal back for the Turks eight minutes into the second half to give them hope for the second leg.
Luuk de Jong scored a hugely fortunate penalty to give FC Twente a 1-0 home leg win over Schalke.
The striker clipped his own heels just outside the area and went down, with defender Joel Matip in close attendance.
But Scottish referee Craig Thompson pointed to the spot and sent off Matip before De Jong smashed the 61st-minute penalty into the roof of the net.
AZ Alkmaar took a 2-0 first leg lead over Udinese thanks to goals from Maarten Martens (63) and Erik Falkenburg (84).
Hannover drew 2-2 at Standard Liege, Mame Diouf scored the equaliser in the 56th minute.
Lars Stindl's penalty put the Germans in front in the 22nd minute, but goals from Yoni Buyens and Mohamed Tchite turned the game around before the break.
Elsewhere, Olympiacos won 1-0 in Ukraine against Metalist Kharkiv.
David Fuster scored the only goal of the game five minutes after the break, and the hosts had Edmar sent off eight minutes from time.