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The Spaniards moved into the quarter-finals of the competition for the first time after Ben Yedder scored with a sharp turn and sweet shot past David De Gea in the 74th minute.
Four minutes later Ben Yedder doubled Sevilla's advantage with a header at the back post which the crossed the line after hitting the underside of the bar despite De Gea's efforts.
Romelu Lukaku pulled one back for Jose Mourinho's side in the 84th but there was to be no comeback for United, who drew 0-0 in the first leg, and the visitors could have added more.
Sevilla have never reached the Champions League quarter-finals but did make the European Cup quarter-finals in 1958.