Ben Yedder struck in the 74th and 78th minutes after coming off the bench two minutes earlier to stun the Red Devils 2-1 at Old Trafford, after the first leg ended goalless.
There was brief hope as Romelu Lukaku netted with six minutes of normal time remaining, but United still needed two more and it was a mountain that Mourinho's side were never likely to scale.
In a game that was fraught with missed chances, a second-half brace by substitute Wissam Ben Yedder was enough to send the Spanish side into the last eight of the competition.
"I sit in this chair twice in the Champions League and knocked United out of the Champions League, with Real or Porto, so I don't think it is something new for the club". We have no time to be sad for more than 24 hours, that's football.
Pablo Sarabia fed Ben Yedder, who drilled an accurate low effort into the bottom right corner to break the deadlock, before adding a second from close range with United rocking.More news: Putin: Jews, Ukrainians 'with Russian citizenship' could be behind United States election meddling
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"The second goal made it impossible".
"It's a big comp, it would be the first time in 60 years for us to get through to this stage".
Mourinho's gamble on Fellaini appeared to have largely backfired as he failed to impose his physical presence on Sevilla's ball players in midfield. "When they scored the first goal the game was completely different". Had David Moyes uttered such comments during his brief reign as United manager, he would have been lambasted, but even Moyes was able to take United into the Champions League quarterfinals.
The visitors were denied a goal within three minutes of the half-time break by a last-ditch Eric Bailly challenge.
"But I believed in myself, believed in the team and we showed we are a great team".
"I am very happy to win here in this special, historic place", head coach Vincenzo Montella said after reaching the quarter-finals.