Paris Saint-Germain, meanwhile, set a record as the quickest team to score four goals in the history of the competition by powering into a 4-0 lead after 17 minutes against Villarreal on their way to a 6-0 win.
Elsewhere, Barcelona beat Aston Villa 3-0 and Bayern Munich won 2-0 at Atletico Madrid.
But the biggest drama of Tuesday night (Wednesday AEDT) came at Old Trafford, where United's troubled season looked set to plunge deeper into crisis after going 2-0 down after 17 minutes against Roma.
It was the quickest United had conceded twice in any European game at their iconic stadium. But a rousing second-half performance included Bruno Fernandes converting a retaken penalty and Matheus Cunha scoring twice before being sent off to seal a much-needed win for under fire coach Michael Carrick.
"I'm not getting too carried away with things," Carrick said. "I think it's an important night. I think it's important for some of the players to have the feeling that they maybe haven't had here before."
The joyous scenes at the end were in stark contrast to earlier, when furious United fans attempted to disrupt the match by hurling tennis balls on the field.
The pre-planned protest was the latest show of dissent from supporters who have long-called for the club's American owners, the Glazer family, to sell their controlling stake.
"We knew after halftime that we still had the fans with us and we felt that at any moment we could change that with a goal, with something. Fortunately for us, we did it today," Fernandes told Amazon Prime.
"Hopefully this win today gives the start of something good for us."
PSG look like the team to beat again as they chase down a hat trick of Champions League titles, which would allow Luis Enrique's team to match the feat of Real Madrid from 2016-18.
Even without the injured Ousmane Dembélé and Khvicha Kvaratskhelia, the French giants had too much firepower for Villarreal, with an early goal blitz effectively putting the game to bed long before halftime at El Madrigal.
Joao Neves, Desire Doue, Senny Mayulu and Fabian Ruiz scored inside the first 17 minutes. UEFA said it was the fastest any team had netted four goals in a Champions League game.
Ruiz and 17-year-old David Boly, four minutes after coming on as a substitute for his competition debut, added goals in the second half for PSG, who are the only team on 12 points.
"Conceding four goals so quickly is difficult for any team," Ruiz said. "You prepare for a match, and then to have four goals scored against you, it's a real blow. We don't look at the scoreboard, we always try to stick to our own style of play and our own rhythm."
There was no goal for newly-crowned Ballon d'Or winner Harry Kane, but Bayern still prevailed at Atletico.
Lennart Karl and Konrad Laimer scored second-half goals for Bayern, who have eight points from four games in the league phase.
Raphinha scored twice for Barcelona to end Villa's perfect start to the Champions League.
Lamine Yamal fired the opening goal at Camp Nou in the seventh and Raphinha put the game beyond doubt with a double after the break.
Sporting Lisbon beat Shakhtar Donetsk 1-0 and Aleksey Batrakov scored two goals in quick succession to seal a 2-0 win for Galatasaray against Stuttgart.
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