The former United midfielder was Tuesday appointed until the end of the season to replace the sacked Ruben Amorim.
Goals from Bryan Mbeumo and Patrick Dorgu secured a morale-boosting victory at Old Trafford against title-chasing City, who remain six points behind Premier League leaders Arsenal.
Pep Guardiola's visitors dominated possession but United were the more threatening team, with seven shots on target to City's one.
The win lifted United to fourth in the table ahead of the later kick-offs on Saturday.
"Unbelievable, Unbelievable," Argentine defender Martinez told Sky Sports. "Being here with the fans, with the team we really know how we suffer during the season.
"But the only way is putting the face here and showing here who we are. Today is the best moment to do it. Not just the players on the pitch, the staff, everyone was part of that.
"I think one important thing that Michael Carrick said was 'Use the energy of the people' and today I think we did it and when we are together like this it is impossible to lose at home."
He added: "Today we defended so good, we were compact all the times. And then you have the confidence to go to the duels. Today you feel a different energy when you see the eyes of the players. Today the team won and that is the most important thing."
Manchester City's Bernardo Silva said United deserved to win.
"They had more energy and intention," he said. "It always felt like they were close to score and we were never there.
"We gave them what they want. They took the game to where they are good and we are not so good.
"Since I arrived here in England we have always been a very controlled team. There was a lot of things missing from us. Definitely a bad day for us."
The Portugal midfielder admitted the gap to Arsenal, who face Nottingham Forest later on Saturday, was daunting.
"It is a big gap, we won't lie and say it isn't a big gap, it is," he said. "We will fight until the end."
Carrick previously had an unbeaten three-game stint as caretaker boss at Old Trafford in 2021.