Second-half substitute Wissam Ben Yedder scored twice in four minutes to send the Spanish underdogs into the quarter-finals of the competition for the first time.
Romelu Lukaku pulled one back for the Red Devils with six minutes remaining, but it was too little, too late for United to rally after 170 minutes of unambitious deadlock.
Despite a confidence-boosting victory over Liverpool on Saturday, the hosts reverted to the cautious approach which restricted them to a goalless draw in Spain last month.
Sevilla continued to create half-chances and the toothlessness of Luis Muriel, who also missed a golden opportunity in the first leg, could have cost them had his replacement not been far more clinical.
Ben Yedder finally broke the stalemate on 74 minutes, only 88 seconds after being called upon off the bench, escaping the attenton of Eric Bailly and rifling into the bottom corner from Pablo Sarabia's pass.
That strike left United needing two, because of their failure to score an away goal, and their task was made even more difficult four minutes later when Ben Yedder turned the ball home from Correa's near-post flick-on. David de Gea pushed the header onto the crossbar, but it dropped down and back over the line.
The Red Devils, who left Old Trafford to boos at full time, managed only four shots on target across both legs, with Sevilla goalkeeper Sergio Rico producing athletic saves from Marouane Fellaini and Jesse Lingard either side of the break on the rare occasions that he was tested.
Jose Mourinho's side briefly threatened another famous European comeback when Lukaku finished emphatically from close-range following a corner, but Chris Smalling and Anthony Martial missed late chances as United's first Champions League knockout campaign in four years came to a disappointing end.
MAN UNITED (4-2-3-1): De Gea; Valencia (Mata, 77'), Bailly, Smalling, Young; Matic, Fellaini (Pogba, 60); Alexis, Lingard (Martial, 77'), Rashford, Lukaku
SEVILLA (4-2-3-1): Rico; Mercado, Kjaer, Lenglet, Escudero; N'Zonzi, Banega, Sarabia, Vazquez (Pizarro, 88'), Correa (Geis, 89'); Muriel (Ben Yedder, 72')