After a turgid first half, Sevilla goalkeeper Sergio Rico kept United out twice with good saves, but the hosts were shaken with 16 minutes remaining as Wissam Ben Yedder netted with a low effort just minutes after coming on.
United needed two goals to progress, but Ben Yedder netted again with a header at the far post moments later, the ball just crossing the line after David de Gea failed to keep it out.
Romelu Lukaku pulled one back from close range with six minutes remaining, but it ended 2-1, with the draw for the last eight on Friday at 11am.
After a goalless first leg, United were lacklustre in the first half as Sevilla failed to take advantage of dangerous situations in the final third.
Romelu Lukaku vies with Sevilla defender Clement Lenglet
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Rico was troubled only once, as Marouane Fellaini's left-footed shot from an angle was well beaten away, in what was a subdued night from the off at Old Trafford.
United again struggled for any rhythm after the break, and fell behind on 74 minutes as Ben Yedder struck from 18 yards into the bottom corner, with his first touches of the ball.
It was 2-0 just three minutes later, as De Gea flapped at Ben Yedder's far post header from a Joaquin Correa cross, the assistant referee confirming the ball had just crossed the line after hitting the bar.
Chris Smalling missed a clear chance at the far post from Marcus Rashford's free-kick, before Lukaku spurned a chance from close range as Sevilla's defence stood firm.
Lukaku pulled one back with six minutes remaining, slamming home Rashford's corner from six yards, but there was still a chance for Sevilla to grab a third, only for Ben Yedder to see his one-on-one effort blocked by the knees of De Gea.
The match-winner, the super sub, and the one man who took the game by the scruff of the neck in the final third.
Ben Yedder made his first goal, with his first touches of the ball, look extremely easy. Finishing low into the bottom right corner with a fierce effort, his one piece of quality all but won the game for Sevilla.
His team-mates, Steven N'Zonzi and Clement Lenglet, also deserve a mention.