City clawed their way back from 2-0 down on the night to win 3-2 against visiting Sporting Lisbon but went out on away goals after a 3-3 aggregate result.
United were outclassed in both legs by Athletic Bilbao, losing 2-1 on the night in Spain and 5-3 overall.
Two down at halftime at Eastlands, City staged a breathtaking rally with two goals by Sergio Aguero and a Mario Balotelli penalty to stand on the verge of a remarkable comeback.
A fairytale ending was millimetres away when goalkeeper Joe Hart's header whizzed just wide in the last minute of stoppage time.
There was no such passion on show by an apathetic United who were undone by a stunning first-half volley from Fernando Llorente and a deflected strike from Oscar de Marcos.
United forward Wayne Rooney scored a superb 80th-minute consolation goal but the damage was already done as the enthusiasm and talent of La Liga's seventh-placed side proved too much for the English Premier League champions.
"I don't think we can complain," United manager Alex Ferguson told Channel Five television. "There are always lessons in every football match whether you win or lose."
Sporting and Bilbao are joined in the last eight by Valencia, who drew 1-1 at PSV Eindhoven to go through 5-3 overall, and Hanover 96 who beat Standard Liege 4-0 for a 6-2 aggregate win.
AZ Alkmaar also progressed despite going down to 10 men after two minutes when Nick Viergever was sent off and falling two goals behind in the first quarter of an hour at Udinese.
The Dutch side eventually lost 2-1 but squeezed through 3-2 on aggregate.
Three La Liga sides advanced with Atletico Madrid joining Bilbao and Valencia in the next round after a 3-0 win at Besiktas gave them a 6-1 aggregate triumph.
Schalke 04 also made the quarter-finals having overhauled a first-leg deficit to beat Twente Enschede 4-1 on the night and 4-2 overall.
Metalist Kharkiv of Ukraine progressed on away goals after a 2-2 aggregate result against Olympiakos Piraeus.
(Writing by Sonia Oxley, editing by Tony Jimenez)
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