Carlos Tevez's first-half penalty ruled out Marouane Fellaini's opener to help City overtake Manchester United, who face Reading in the evening kick-off, at the top of the table.
Everton, who were the last team to beat Manchester City on their home ground, travelled to the Etihad having won five of the last six Premier League meetings between the two sides.
The Toffees took the lead after 33 minutes when Fellaini bundled Leighton Baines's free-kick over the line after his initial header was saved by Tim Howard, his eighth goal of the season.
Tevez almost pulled City level moments later when his goal-bound header was saved by Howard before Edin Dzeko's half volley was turned around the near post by the Everton stopper.
Fellaini conceded a penalty from the resulting corner as he brought Dzeko to ground and Tevez made no mistake from 12-yards, placing the spot-kick down the centre of the goal.
City dominated the second-half but, for all of their possession, failed to create chances. Maicon forced the only save out of Howard with a powerful drive from 18-yards while Sergio Aguero and Mario Balotelli failed to make an impact off the bench.
Nikica Jelavic almost caught Joe Hart off-guard with an awkward late free-kick but Everton were more than happy to take a point back to Merseyside and remain sixth in the Premier League.
Manchester City are now 21 games unbeaten in the league and will equal their all time record set in 1936-37 if they avoid defeat in the Manchester derby next Sunday.