The Blues dominated possession and produced four superb saves from the Everton keeper - two of them penalties - but almost came unstuck thanks to a goal on the break
The £14 million signing of Nolito set few pulses racing among Manchester City fans this season but he has already proven his worth to Pep Guardiola's side.
On a day when City's dedication to playing the percentages so very nearly backfired, Nolito showed exactly why Guardiola wanted him as part of his transitional season at the Etihad Stadium.
Guardiola will no doubt be livid after this game. He was magnanimous after the 2-0 defeat at Tottenham, holding his hands up and admitting Spurs were the better side.
Here, though, there was something missing, even if luck deserted them in a way that would have Football Manager players throwing their laptops out of the window.
Everton keeper Maarten Stekelenburg saved penalties from Kevin De Bruyne and Sergio Aguero, the most reliable City stalwarts, and also then kept both men out with two save-of-the-season contenders.
If the Dutchman's heroics were not enough, Everton threatened to win the match with the definition of a smash-and-grab goal on the break.
Guardiola has dismissed countless questions over whether he'll change his possession-oriented style since arriving in Manchester but he would have had to face a couple more after Romelu Lukaku raced forward one-on-one with Gael Clichy and produced a consummate finish to give the visitors the lead.
City had dominated the game up until that point but could not find a way through Everton's massed ranks. If Spurs and Celtic's demanding high-energy game is above what most of City's Premier League rivals can produce, perhaps the Toffees' defensive approach is more attainable.
They sat back with 10 men behind the ball and tried to counter-attack through three attacking players, matched up against Guardiola's three-man defence.
City should get used to that approach given it restricted them to just a point, but in reality Everton rode their luck just as much as City did and will surely fare better against other opponents who deploy similar spoiling tactics.
What will annoy Guardiola most is that the home side were often their own worst enemy, misplacing countless passes that had their coach slapping his hands on his thighs in frustration on several occasions.
City will feel aggrieved that, despite Everton's resilience, they have not won this game. After bossing possession, winning two penalties out of Everton's mass of bodies and producing two magnificent saves from Stekelenburg, they did more than enough to take all three points.
As it was, they were left to thank quiet man Nolito. His signing did not create the same kind of buzz that welcomed Ilkay Gundogan, John Stones or even young prospect Leroy Sane, but he has done exactly what he needed to do after leaving Celta and, according to the man himself, snubbing Barcelona.
City could not sign the two ball-playing centre-backs they craved this summer and have had to make do with Aleksandar Kolarov and Nicolas Otamendi, who you sense does not have Guardiola's full confidence, alongside Stones. They have paid the price for that on a couple of occasions - notably with Kolarov's horror shows at Celtic and Spurs, but they have had few issues in the attacking midfield department.
That is because Nolito knows exactly what Guardiola wants and, at least nine times out of 10, he does it. He is not as exciting as Sane and he is not as smooth as Gundogan or Stones but he gets the job done.
So often he is in the right place at the right time, and so often when his team-mates are struggling to find space, as they did here today.
It was he who got free to nod in a leveller that will just about stave off more questions about Guardiola's philosophy.
Word is that Barcelona's board were not as keen to bring Nolito back to the club as coach Luis Enrique, who had been trying, with no success, to persuade his board to part with the necessary funds to sign the Spain international for 18 months before City swooped and brought him to Manchester.
He left Camp Nou in the first place to go out and show what he can do and, after good spells with Benfica and Celta, he has finally got his shot at the big time.
He has been quietly impressive so far and once again showed that he is worth just as much fanfare as City's other summer acquisitions.
This was a day when not everything went City's way, but they have something to show for it - and it is Nolito, on his 30th birthday, who once again came up with the goods.