Sergio Aguero scored twice, with Yaya Toure and Samir Nasri grabbing the others as City romped to their biggest home win over their neighbours since 2004.
Blues fans revelled in the triumph, cheekily chanting "One David Moyes" and "Getting sacked in the morning" before the end, as they targeted United's new manager.
And Pellegrini, also making his debut on the touchline in a Manchester derby, admitted it could be a significant victory.
"It is very big," said the Chilean.
"Always winning against Manchester United and the way we won is important.
"You don't expect to win 4-1 against United. Normally derbies are more equal.
"But today we had a very good day and that allows us to have a superiority against Manchester United.
"The fans will enjoy this for the next three months.
"For the team, winning the way we did today will give a lot of trust.
"Winning a derby will also give them a lot of confidence."
It seemed churlish to take anything away from City but United were as poor as they have been at any time against their rivals in recent years - even in the 6-1 demolition two seasons ago when the Red Devils were handicapped by the dismissal of Jonny Evans and conceded three times in stoppage time.
"Maybe they couldn't play better because we played very well," said Pellegrini.
"We tried to create space and did it.
"We didn't give them space to damage our goal."
City are now in third place in the Barclays Premier League, their margin of victory allowing them to ease ahead of Chelsea and Liverpool on goal difference.
And Pellegrini is hopeful of more memorable days to come.
"We must improve every day," he said. "Not all the games are the same. I think we are just starting another style and another way to play."
City captain Vincent Kompany was keen to play down talk of the team making a "statement" in the title race.
"I think you make a statement over the whole season," he said. "It's too early to say it's a defining game.
"When we won 6-1 at Old Trafford I said we needed to stay humble, it's only three points.
"But it gives us all the confidence and allows fans to celebrate for a couple of days."
Asked about why his side were so dominant, Kompany told Sky Sports 1: "Maybe the game meant a little bit more to us than to them.
"We've been looking forward to it, our fans have been speaking about it for a long time and we needed to win.
"We have a fantastic home record and we know there is no reason we should fear the opposition."