Manuel Pellegrini's side dominated the second half but were repeatedly denied by away keeper Tim Howard.
It was a supreme defensive display from the Toffees, who had been the better attacking side in the first half.
They also rode their luck, as City were denied a late penalty for a foul on Raheem Sterling by John Stones.
For Everton, it is a point won - and a confidence-boosting one, following a run of seven games in which they have won just once.
But it is two key points dropped for City and they remain three points behind joint league leaders Arsenal, held 3-3 by Liverpool, and Leicester, 1-0 winners at Tottenham.
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The game began in a half-full Etihad Stadium as traffic problems prevented many fans arriving by kick-off.
However, you would forgive many of the City supporters who did eventually make it to the ground for wishing they had not. Before the break, the home side looked sluggish and were outworked by a dynamic away side.
They did produce a couple of chances - the best of them a Yaya Toure header from eight yards that was palmed away by Howard - but could easily have trailed.
They improved in the second half as Sergio Aguero became more involved, but still struggled to break their opponents down, even after the introduction of David Silva with 15 minutes to go.
The game would end in desperate, vain calls for a penalty for Stones' admittedly risky sliding challenge on Sterling.
City boss Manuel Pellegrini was adamant that Stones' challenge should have been penalised and his side awarded a "clear penalty".
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"Everyone in the stadium could see it was a penalty - it was so near to the referee and so clear," said the Chilean. "Both Raheem's legs were taken by Stones but if he doesn't whistle we can't say anything."
Unsurprisingly, Everton boss Roberto Martinez had an opposing view of the incident and praised the "maturity" of referee Roger East.
"We have been on the back end of receiving a penalty, in the last seconds against Stoke," said the Spaniard.
"Stones goes to slide to block the ball, Sterling does not play the ball and waited. In my eyes it is an area that is not certain if it's a penalty or not.
"I believe the referees should control their emotions and the referee did exactly that. It is impossible to be 100% certain and I am pleased he did this and showed his maturity."
This game was the second part of a fixture trilogy between these sides in January, falling between the two legs of a Capital One Cup semi-final.
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A week ago, Romelu Lukaku's winner gave Everton a 2-1 victory at Goodison Park, and they impressed again in a performance that showcased both sides of their game.
In the first half, the Toffees were patient and precise when they had the ball and hard-working when they did not. But for some wayward shooting, they would have led at the break.
Gerard Deulofeu and Lukaku were both denied with shots from an angle in the box, while the latter set up Leon Osman for the best chance - a volley from 15 yards that flew just past Joe Hart's far post.
After the break, Roberto Martinez's side were firmly on the back foot and spent long spells without the ball but demonstrated discipline and commitment to limit City to only a few goalscoring chances.
Manchester City are at home again on Saturday, with Crystal Palace the visitors.
Everton face another tough away trip on the same day, as they travel to Stamford Bridge to face Chelsea.
Manchester City missed the chance to make up ground in the Premier League title race as they were held to a goalless draw by a stubborn Everton.