Josko Gvardiol's header put City on course for a much-needed win at the Etihad on Sunday but United captain Bruno Fernandes levelled from the penalty spot in the 88th minute.
Amad Diallo won that penalty and then conjured up a memorable 90th-minute winner as Amorim celebrated his biggest win since replacing the sacked Erik ten Hag last month.
"It's been a difficult start to the season for us and not much for our fans to cheer about," said defender Maguire.
"It was a great performance in the end to come here and have 50-50 of the ball, have the same amount of shots as them.
"It's not been like that in recent years, so it's an improvement but still a long way to go."
United suffered Premier League defeats to Arsenal and Nottingham Forest earlier this month and are still a lowly 13th in the table after a poor campaign so far.
"Obviously, we had back-to-back defeats in the Premier League, so to come here and to put a statement down is really important for our confidence and the belief of where we're going and where Ruben's going to take us," said Maguire.
"I'm sure it's going to be a positive journey."
United's victory meant City's wretched run stretched to eight defeats in 11 matches in all competitions.
Two of those losses have been overseen by Amorim, who led Sporting Lisbon to a stunning 4-1 Champions League win in his penultimate match before moving to Manchester.
"I think you can see we have a good structure, a good style of play, the way that he wants us to work," Maguire said of the new manager's impact.
"But, yeah, I think his demands, he's tactically very good. He's proven that he's a winner."
