Substitute Wissam Ben Yedder was the match-winner for the Spanish side, shooting from inside the area just two minutes after coming on and heading in at the back post from a corner.
After posing little threat in the goalless first leg, United again looked second best against the La Liga side, though Marouane Fellaini's thumping effort did draw Sergio Rico into a save and the away goalkeeper kept out Jesse Lingard's low strike.
Romelu Lukaku lashed home from close range with seven minutes remaining, but it proved to be merely a consolation.
United exit the competition along with Tottenham, but Premier League leaders Manchester City and Liverpool progressed to Friday's quarter-final draw. Chelsea can join them if they beat Barcelona on Wednesday.
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