AS Roma actually led 2-1 at half-time after strikes from Lorenzo Pellegrini and ex-Manchester City marksman Edin Dzeko but Manchester United hit back emphatically in a stunning second-half.
Bruno Fernandes finished with two goals and two assists, while Edinson Cavani netted a double with Paul Pogba and Mason Greenwood also on the scoresheet to confirm a major advantage ahead of the second leg in Rome.
“No, you don’t feel the job is done but it was a good job done,” United boss Ole Gunnar Solskjaer told BT Sport. “I think the character showed that they came back and didn’t lose their heads.”
He continued: “You could say we’ve got one foot [in the final], but you need two feet. We’re still halfway.
“A fantastic result to build on but you’ve got to be focused. I’ve seen stranger things.
“Football comes back to bite you if you take it too easily. We’ve just got to focus on Liverpool first and then Roma.”
United’s probable final opponents are no clearer after 10-man Arsenal rallied after trailing 2-0 early to Villarreal in Spain, to finish with a 2-1 defeat.
Manu Trigueros and Raul Albiol had earned the Yellow Submarine a two-goal lead within half an hour but the Gunners showed mettle to stay in the tie, despite Dani Ceballos’ 57th minute sending off.
Nicolas Pepe scored a 73rd minute spotkick before Etienne Capoue also saw red for Villarreal with Pierre-Emerick Aubameyang coming close to a 93rd minute leveler.
Arsenal boss Mikel Arteta said: “Obviously we are disappointed that we lost the game, because it’s not the result that we wanted to take from the game.
“But after the story of the game, I think we have to take it because in the tie we are alive and in the circumstances that developed during the game, it put the game and the tie in real danger at some stages.”