Ike Ugbo continued his superb run in the competition after firing in his eighth Youth Cup goal of the campaign in the 44th minute.
Callum Hudson-Odoi did the hard work for the opener after seeing his effort rebound off the post before he set up Ugbo for the finish.
But City reacted quickly and after a strong finish to the first half, they were rewarded in the 62nd minute when Phil Foden restored parity with his effort after the hosts had robbed Chelsea of the ball in midfield.
Neither side could find a winner in the dying stages of the game, which means there is everything still left to play for when the second leg is contested at Stamford Bridge on 26 April.
Chelsea's Under-18s side arrived to Manchester City's Academy Stadium for the first leg in high spirits after booking their place in their eighth final in 10 years with a 9-2 aggregate thrashing of London rivals Tottenham.
Such has been their success in the FA Youth Cup that the Blues have triumphed four times in five seasons - with the last two victories coming at the expense of City.
And perhaps mindful of those two disappointments, the hosts made Chelsea work hard in the early stages of the first leg with the visitors taking 15 minutes to settle into the contest.
After finally doing so, however, Chelsea came close to scoring first when Hudson-Odoi forced a save from City goalkeeper Arijanet Muric.
The hosts - who were being closely watched by City's first-team manager Pep Guardiola - took little time to react and they should have scored when Brahim Diaz launched a counter-attack and delivered a superb through-ball to striker Lukas Nmecha.
But the Germany-born forward drove his resulting low shot agonisingly wide of the far post before he combined with Diaz minutes later to threaten Chelsea's goal once again.
After being played in by Diaz, Nmecha did well to round the Blues' centre-backs but, despite having space and time, the striker blazed the ball over the bar - which resulted in Guardiola jumping out of his seat and airing his frustrations in front of the television cameras.
The miss proved costly as Chelsea then followed up two subsequent near misses with a goal.
Hudson-Odoi created the opener when he ran at the City defence and unleashed a low shot that hit the post and rebounded back out to him.
Noticing Ugbo nearby, he cut the ball back for the Chelsea striker who coolly drove in to hand the Blues the lead.
The goal injected life into the contest, and City responded through Jadon Sancho, who fired wide seconds before the half-time break.
The hosts continued to push men forwards in the second period, and they finally rewarded when Sancho robbed Chelsea of the ball before passing to Foden, who fired in a low effort just after the hour mark to leave the tie finely poised ahead of the second leg next week.