Matheus Cunha was the hero for Michael Carrick's side, scoring twice in the closing stages after Bruno Fernandes had converted a penalty to inspire United's comeback from a two-goal deficit.
Roma made a dream start, taking the lead in the 11th minute through Matías Soulé, who finished after receiving a pass from Rodrigo Mora.
The visitors doubled their advantage six minutes later when Donyell Malen converted from the penalty spot after Ayden Heaven was penalised for handball inside the area.
United struggled to respond during the first half, with Roma threatening to extend their advantage. The home supporters expressed their frustration as the players headed into the dressing room trailing 2-0.
The situation became even more difficult for United shortly after the restart when Carlos Baleba was forced off with a hamstring injury.
However, the hosts gradually found their rhythm and were handed an opportunity to reduce the deficit when Cunha was brought down inside the penalty area by Gianluca Mancini.
Bruno Fernandes initially saw his penalty saved by Roma goalkeeper Devis Vásquez, but the Portuguese midfielder was given another opportunity after a Roma player encroached into the penalty area.
The United captain made no mistake with his second attempt, firing into the top-left corner in the 58th minute to make it 2-1.
The goal transformed the atmosphere at Old Trafford, with United pushing aggressively for an equaliser. Bryan Mbeumo came agonisingly close when his effort struck both posts before bouncing away.
United continued to apply pressure and finally drew level in the 83rd minute. Fernandes delivered a dangerous ball into the penalty area, and after Leonardo Balerdi failed to deal with it, Cunha calmly finished beyond Vásquez.
The Brazilian completed the turnaround just three minutes later, collecting a brilliant pass from substitute Shea Lacey before driving forward and firing past the goalkeeper at his near post.
However, Cunha's night ended controversially when he received a second yellow card in the 90th minute for a late challenge on Balerdi, having already been booked for celebrating his goal in the crowd.
Despite finishing with 10 men, United defended their advantage during the closing stages to secure a memorable 3-2 victory.
United end difficult run with dramatic victory
The result represents a major boost for Manchester United and manager Michael Carrick, who had been under pressure following a disappointing run of results.
The victory also gives the English side renewed hope of progressing in the Champions League league phase.
For Roma, the defeat will be difficult to accept after they controlled much of the first half and appeared comfortable with their two-goal advantage.
United will now turn their attention to their next Premier League fixture against Aston Villa at Old Trafford on Sunday.
Full-time result:
• Manchester United: 3 (Bruno Fernandes 58' pen, Matheus Cunha 83', 86')
• AS Roma: 2 (Matías Soulé 11', Donyell Malen 17' pen)
Red card: Matheus Cunha (90').