Rico Lewis' first-half penalty had put us ahead after Dire Mebude was brought down in the area.
But Everton striker Martin Sherif levelled in the latter stages to ensure the points were shared on Merseyside.
Ben Wilkinson’s side dominated proceedings from early on, keeping possession well and giving the home side little respite with an intense high press.
Mebude's pace caused Everton plenty of problems down the right-hand side and the winger nearly opened the scoring when he had a tight-angled shot saved from inside the box.
But the winger wasn't to be denied a second time, as he then provided us with our opportunity to open the scoring, being hauled down inside the area by Elijah Campbell.
Captain Lewis then stepped up to put us ahead, sending keeper Jak Stewart the wrong way for his second goal of the season, adding to his solo effort against Manchester United last week.
Lewis then had to be on hand to prevent a goal at the other end, as he cleared an Everton header off the line from a corner, ensuring City went in ahead at the break.
Into the second half, our young blues kept up the same intensity that they showed in the first, seeing Everton continuously pinned back by their own box.
And we nearly doubled our lead when Mebude once again broke down the right, losing defender Charlie Wilson, driving into the box and firing a low shot goalward which was saved by Toffees’ keeper Stewart.
But despite us having more of the chances, the home side levelled in the 76th minute when striker Sherif converted his left-footed strike from inside our box.
City were thankful to our stopper Josh McNamara late on, when he saved brilliantly from Issac Heath's close-range effort, before a follow-up effort was cleared off the line by Jadel Katongo to preserve a point for Wilkinson's side.
How we lined up
McNamara was named between the sticks, with Lewis, Issac Smith, Katongo and Matty Whittingham making up a four-man defence.
Mo Susoho provided additional defensive cover in the holding midfield role, with Kian Breckin and Nico O’Reilly in more advanced roles.
The attacking trio compromised of Mebude and Joel Ndala on the flanks, with Josh Adam leading the line for City at Finch Farm.
What it means
Ben Wilkinson’s side continues our unbeaten start to the season, making it two wins from two following our victory over Manchester United last week.
This sees us move up to third in the standings at this early stage of the campaign, tied with both Wolves and Burnley.
Our Young Blues will next take on Nottingham Forest in the U18s Premier League on Saturday 28th August at the Academy Stadium.
News about Everton v Man City
Guardiola: Gabriel Jesus deserves the best
Pep Guardiola was delighted to see Gabriel Jesus earning the praise he deserves, following Saturday's 5-0 triumph over Norwich City. Watch more.