Maarten Stekelenburg saved two penalties for Everton as his side escaped with a point against Manchester City at Etihad Stadium.
Kevin de Bruyne was the first to miss in the opening period, seeing the Dutch goalkeeper push his effort away, and Sergio Aguero saw the same outcome in the second half.
Despite dominating possession, City went behind as Romelu Lukaku opened the scoring with a cool finish.
Nolito headed in for the hosts from David Silva's inch-perfect cross, but De Bruyne had a drive brilliantly saved by Stekelenburg, as City failed to find the winner.
City starting to stumble
Following their perfect start to the season when they won their opening six games, City have stumbled in their past two with defeat at Tottenham and this draw against Everton.
Boss Pep Guardiola stood scratching his head on the touchline in the second half, during a game in which his side had 19 shots and 72% possession but only found the net once.
Leroy Sane was given his first Premier League start for City and the German international showed his promise with an impressive performance, while Raheem Sterling also threatened with his pace.
But it was the host's philosophy of trying to work the ball into the net which let them down, finding a resolute Everton defence that sat deep to snuff out wave after wave of opposition attack.
Having somehow fallen behind, City did manage to grab an equaliser through substitute Nolito's header from close range.
Williams a rock for Toffees
Ronald Koeman's side could have broken into the top-four with victory, and took the lead against the run of play through Lukaku, who ran clear from the halfway line and slid home the opener.
Team-mate Ashley Williams showed his steeliness at the back by making 13 clearances - more than any other player on the pitch.
But defensive partner Phil Jagielka had a shocker, conceding both penalties after tripping Silva by sticking his leg out and a hack on Aguero inside the area.
Everton had goalkeeper Stekelenburg to thank for keeping out two spot-kicks, as well as further excellent stops from Aguero - who started on the bench - and De Bruyne.
The Belgian, in his attempts to get City on the front foot, lost possession 32 times in the match.
Mixed day for Stones
Centre-back John Stones made 77 league appearances for Everton before joining City in the summer for a huge £47.5m.
Although he is now learning to bring the ball forward at the right time and clearing it to safety when required, there is still work to do on the Englishman's game.
For Everton's goal, he found himself beyond the halfway line, failing to win the ball as Bolasie intelligently nipped in front to nick the ball.
It left City with a gaping hole in defence, and Lukaku against Gael Clichy was no contest as the Belgian shrugged past the Frenchman to score.
Man of the match - Maarten Stekelenburg (Everton)
Analysis
Former England defender Martin Keown on BBC Radio 5 live:
"It was a good, honest performance from Everton and they deserve this point today. For all Manchester City's dominance you have to praise what has been a gallant performance by Everton.
"You look at the two managers and Ronald Koeman will be happier than Pep Guardiola. City dominate and play football of the highest quality but just missed that little bit in the final third today."
What they said:
Manchester City manager Pep Guardiola: "We did absolutely everything. Everton created just one chance which was a goal. I'm a little bit sad and disappointed for the players. But it's a big compliment to Everton and Ronald Koeman so I say to them 'well done'."
Everton manager Ronald Koeman: "I am satisfied with a point because not many teams leave this stadium with a point. You need to be a little bit lucky and a goalkeeper to have the match of his life, and you need a work-rate from your team. A big compliment to the team because we fight."
What next?
Everton travel to Burnley in the league on Saturday, 22 October (kick-off 15:00 BST), while City host Southampton the next day (kick-off 13:30).
The stats you need to know:
- Ronald Koeman has earned the same number of points (15) as Roberto Martinez did in his first eight Premier League games in charge.
- Manchester City are unbeaten in their past seven Premier League meetings with Everton (won four, drawn three).
- Romelu Lukaku has scored six goals in his last five Premier League games after a run of 11 consecutive appearances in the competition without a goal.
- City have failed to score four of their eight penalties in all competitions this season.
- David Silva has provided more Premier League assists than any other player in the competition since his debut (59).
- Phil Jagielka is the first Premier League player to concede two penalties in one game since Fernando Amorebieta (Fulham v Man City, March 2014).
- Maarten Stekelenburg saved two penalties against Man City; the eighth occasion a goalkeeper has done so in a single Premier League game.