Edinson Cavani and Bruno Fernandes scored twice but it was the late goals from Paul Pogba and Mason Greenwood that will have Ole Gunnar Solskjaer confident his side are on the brink of reaching the final.
Roma had led 2-1 at half-time after goals from Lorenzo Pellegrini and Edin Dzeko but now need a miracle to progress.
United had lost their last four semi-finals under Solskjaer, including the tie against Sevilla in this competition last year. But Cavani and Fernandes were on fine form to leave United fans dreaming of a final against Villarreal or Arsenal.
We have reviewed some of the key performers from a memorable night at Old Trafford.
✔️ Paul Pogba: 7.3
It was a fine Pogba header from Fernandes' cross that gave United their fifth goal of the night with 15 minutes left.
It was the outcome Pogba deserved after a good display, even though his handball had conceded the penalty from which Pellegrini scored early on.
That decision went against him shortly after he had a hand in United's opener, his fine solo run leading to Cavani impressively setting up Fernandes.
Since returning from injury six weeks ago, Pogba has scored two goals - his other being a priceless strike against Milan in this competition - and added five assists.
⭐️ Edinson Cavani: 10
When Man Utd signed Cavani on a free transfer they hoped his big-game experience would come through when it mattered.
That was certainly the case here, as he produced a superb display in attack to finish with two goals, two assists and a perfect 10 rating.
Cavani racked up seven shots on goal, four of which were on target, both game highs.
His pass for the sixth United goal from Greenwood four minutes from time was sublime, while he had also combined with Pogba to create Fernandes' opener.
Not captured by the stat sheet, he also won the controversial penalty when he was deemed to have been fouled by ex-United defender Chris Smalling. Fernandes scored it to put United 4-2 ahead.
⭐️ Bruno Fernandes: 10
He clipped a cool finish past Pau Lopez to set the tone after nine minutes and later converted the crucial penalty.
Fernandes' precise cross was headed in by Pogba while he also set up Cavani for a powerful finish three minutes after half-time, a goal which sparked a fantastic second half for the Red Devils.
The playmaker ended the game with six chances created and was back to his best, emphatically halting what had been a quiet four-match spell by his standards where he had not produced a goal involvement.