Maguire says the fans spurred Ole Gunnar Solskjaer’s men to a 3-2 victory over the Italians at Old Trafford.
It was all gloomy and funereal at the Theater of Dreams when the visitors established an early dominance thanks to goals from former Chelsea midfielder Mario Pasalic and Merih Demiral.
England forward Marcus Rashford handed Manchester United a lifeline eight minutes after the break when he pulled one back.
Maguire got the equalizer on 75 minutes before Cristiano Ronaldo delivered the coup de grâce on the 81st minute mark to secure the maximum points for the English side.
"The first half we conceded two bad goals. Two big moments in the game. For the first they caught us out and then the second it is a set-play that we shouldn't be conceding,” Maguire said after the game.
“We created a lot of chances in the first half so we knew at half-time that the next goal would be crucial.
“We got the goals we needed and deserved and I think we deserved the win in the end. Yes, we made it difficult but it is a tremendous victory in the end.
"I had eyes on it all the way [Cristiano Ronaldo's goal]. I was right behind him. His leap, his timing, it was perfect right in the corner.
“We see it day in, day out in training and in the goals he has scored throughout his career. He has come up with a massive goal for us again in the Champions League.
"It is normally with my head [that he scores]. I have not manged to put one away. It is my first goal for the season and the Champions league. A proud moment with my family here.
"We were two goals down and we need to sharpen up with that. But we had plenty of chances.
“I think Marcus [Rashford] had two one-on-ones in the first-half. We are working hard to be more solid but it is about winning and the last three or four games we have conceded too many set-pieces.
“It is fine margins but we need to do better. We showed great togetherness. We were clapped off at half-time by some of these fans and they gave us the belief we needed."