AS Roma looked in control at the interval after Matías Soulé opened the scoring in the 11th minute and Donyell Malen converted a penalty six minutes later.
Michael Carrick's side had struggled to contain the visitors, but the United head coach insisted he had seen enough before the break to believe a recovery was possible.
“I thought we grew into it in the first half, actually, after giving the goals away too easily,” Carrick told Amazon Prime.
“We spoke at half-time about how this place can change, and it only takes one moment. The boys made the most of every little opportunity, really, and were fantastic in the end.”
Bruno Fernandes gave United a route back into the contest when he converted a penalty in the 58th minute, after his initial effort was saved and ordered to be retaken because of encroachment.
Carrick said he had tried to keep his players positive against a Roma side who pressed aggressively and defended man-for-man.
“I knew we were doing some good things in the game against a really good team,” he said. “Playing against a team that goes man-for-man as much as they do, you probably get more failure than success. It’s about still being positive and the boys responded so well in the second half.”
Matheus Cunha levelled in the 83rd minute after Roma failed to clear their lines, then completed the turnaround three minutes later with a near-post finish from Shea Lacey's pass. Cunha was sent off in stoppage time, but United held on for their first victory in 10 matches.
Carrick was particularly pleased with Lacey's contribution, highlighting the teenager's willingness to work defensively as well as provide the decisive assist.
“He’s so talented and desperate to do well,” Carrick said. “The boys respect Shea a lot. He’s desperate to do well and he was even defending at right-back towards the end.”
Roma head coach Gian Piero Gasperini admitted his side had failed to respond after Fernandes' penalty changed the momentum.
“The first goal for them was a penalty, the second came from an injury and when the match had become a little more balanced,” he told Sky Sport Italia.
Gasperini felt Roma had lost their ability to compete physically as United pushed forward.
“We had to play differently and use the long balls as well, but we couldn't hold their references,” he said. “Perhaps we needed to respond with a few more physical challenges. We suffered from their momentum, amplified by the penalty and the 2-2 goal.”
The Roma coach added that the defeat should provide lessons for his team, while Carrick urged Manchester United to build on a night that restored belief at Old Trafford.