Manchester City were reduced to ten men after just 23 minutes when Phil Foden was sent off, leaving Enzo Maresca's side facing a mountain to climb in the 199th meeting between the two clubs.
Despite the setback, defensive solidarity and a clinical Erling Haaland strike ensured City maintained their 100 per cent winning start to the season with a 1-0 victory over Michael Carrick's side.
Summer signing Anderson was a standout performer in midfield and was named Player of the Match for a tireless display, but he was quick to spread the credit across the squad.
"We are now, kings of Manchester. We always knew it would be a tough game," Anderson said. "Losing Phil at the start wasn't good for us but we said at half time it would make us feel better to win. We took the three points.
"It gives you more motivation to go and get the points. It was really tough for us. We had to run more and fight more. We all shone to a man and we were brilliant."
The midfielder, who joined City from Nottingham Forest for £116 million over the summer, said the physical demands of the fixture suited his and Enzo Fernandez's style in the middle of the park.
"All of our midfield do relish these games. Me and Enzo love to fight, tackle and get on the ball and play. We gave it our all today and everyone was brilliant."
It was Anderson's first taste of the Manchester derby, and he expressed his delight at marking the occasion with victory, having previously described City as the "kings of Manchester" before the game.
"It was my first Manchester derby and to start it like that is brilliant," he said. "It's a new era here and starting here with a win in this game is amazing.
"We're all really happy."
City sit level at the top of the Premier League table having won every match so far this season, with Anderson's performance further underlining the impact of his big-money summer move.