Two goals from Sergio Aguero and tallies from Yaya Toure and Samir Nasri gave City a memorable result over their neighbours at the Etihad Stadium with Kompany praising the calmness of the performance.
"Maybe the game meant a bit more to us than to them," Kompany told Sky Sports.
"We've been looking forward to this fixture, and our fans have been speaking about it for a long time - and we need to win those kind of games.
"We've got an unbelievable home record and there's no reason why we should fear the opposition, even though they are very good.
"We did play with two very good strikers. Defensively, we've had a very good record for the past three years, so we knew that we could attack and we could defend. Mainly I think we stayed so calm."