The Belgian praised the dominant display from his side but warned that the importance of the win would only be felt if the club maintained the form over the season.
City's early spell of pressure resulted in Sergio Aguero's goal after 16 minutes, while Yaya Toure doubled the hosts' lead on the stroke of halftime. Aguero netted his second of the match two minutes after the restart, while Samir Nasri rounded off the win for Manuel Pellegrini's men.
Wayne Rooney netted a late free kick to grab a consolation goal for the visitors, but David Moyes' side had struggled to create clear-cut chances throughout the match.
"Maybe the game meant a little bit more to us than to them. We've been looking forward to this fixture - our fans have been speaking about it for a long time and we need to win those kind of games," Kompany told Sky Sports. “We've got an unbelievable home record, and it was not reason why today we should fear the opposition although they are very good.
"We played with two very good strikers, and I think that defensively we've had a very good record for the past few years so we knew that we could attack and we could defend.
“Mainly we stayed so calm - at 2-0 sometimes we had to absorb a little bit, but we remained calm."
Kompany added that the importance of the result would only be felt if Manuel Pellegrini’s side built on the result over the course of the league campaign.
“You make a statement over the whole season,” Kompany said. “It's too early to say it's a defining game, but it would be silly to give away those very precious three points, and we need to win the next games and then we'll enjoy what we've done today.
“When we won 6-1 at Old Trafford I said we need to stay humble, and it's only three points [today], there's so many important games to win but it gives us all the confidence and it allows the fans to celebrate for a couple of days."