Two quick strikes from Wissam Ben Yedder in the second-half put the Spaniards in control of the tie.
Romelu Lukaku pulled a goal back for the Red Devils on 84 minutes but it was not enough for United, who have not made it to the quarter-final stage of the competition since 2014.
Meanwhile, it was a glorious night for Sevilla, who make their first ever Champions League quarter-finals and they also registered their first ever win in England.
So what went wrong for Jose Mourinho's men? Click the right arrow above to see talkSPORT's Manchester United player ratings...