But City, who've now gone three games without a win in all competitions, equalised in the 72nd minute when Nolito headed in Silva's cross as former Barcelona team-mates turned managers Pep Guardiola (City) and Ronald Koeman (Everton) had to settle for a point apiece.
There are seven players in the final year of their contracts at the Etihad and Guardiola is trying to manage an evolution rather than revolution.
"For the last decade, the last 50 years, Barcelona have dominated football matches", Guardiola said. Off the field, director of football Txiki Begiristain held an equivalent role at Barca and was also a player there, while chief executive Ferran Soriano also had a senior position.
He said: "They are the best team I have played against in my managerial career".
Sure, Everton scrapped to a deserved point here last season under Roberto Martinez but this time the Blues faced a sterner test against a Pep-propelled Manchester City - and so this performance was indicative of just how far they have come under the Dutch manager. When you play like Saturday then it is not a good result.
"We came with two game plans", Koeman said afterward.
"They have three awesome players - Lionel Messi, Luis Suarez and Neymar - up front".
He added: "Of course here there are more games to play [in the Premier League], but I think the intensity in Italy and other places is quite the same".
Enrique:"I've still not spoken with Pep but this was the ideal way to go into the match against City, winning against Depor".
De Bruyne replaced Aguero as penalty-taker and his kick was a nice height for Stekelenburg, who parried the ball away diving to his left. We've made a good start, we want to continue.
"I'm pleased with the point".
"Maybe we didn't deserve one, counting the number of chances, and the fact that there were two penalties missed but football is unpredictable and the key to get a point was the belief and the work rate of the players". However, it was his stops from Aguero and De Bruyne in open play that impressed the most, one with each hand. All we can hope is that he has a bad day and our defence plays the game of its life. Hopefully now they will have all moved on from the doom and gloom feel that hung over the club for a week or two after the Bournemouth and Norwich games.