Edin Dzeko notched a brace as City secured a gripping 3-2 win to edge ahead of their opponents' local rivals Liverpool on goal difference at the top of the table.
An Everton victory would have left Liverpool well placed to secure a first championship for 24 years, leading to suggestions of mixed emotions in the Goodison Park stands on Saturday.
"It was hard and very intense as well - I was thinking that the Everton players have got a lot of friends at Liverpool," Kompany joked when interviewed by Sky Sports after the match.
"It was tough and we ultimately got a great result."
Sergio Aguero cancelled out Ross Barkley's superb opening strike to set City on their way to ending a run of four straight defeats at Goodison - a streak their captain attributed to Everton's competitive spirit and a lack of luck.
"To be honest we've not had a lot of great results but we've never really had too many bad games either," Kompany said.
"It's just one of those places. When we take the lead it's never finished - they always find a way to make it difficult and they always find a way to score a goal that put them back in the game.
"Today was the same. I thought until the last minute both teams had to keep giving everything they had to keep in the game."
City have passed up a handful chances to take a firm hold on the title race since February's 1-0 home loss to Chelsea, seemingly in the most damaging fashion when they slipped to a 3-2 defeat at Liverpool three weeks ago.
He added: "We've had a lot of tough times during the season but I think that we've had a lot of wonderful games.
"The most important thing is to pick up the form at the right time and if we can carry on like this in the next two games, hopefully it should be enough to win those two games.
"I guess the past will tell us to be careful."
