City went behind after 35 minutes when Harvey Vale struck to put the home side ahead, but a stunning long-range strike from Kian Breckin brought us level just before half-time.
Oscar Bobb smashed home from a tight angle just after the break to put the Blues ahead, before Lewis Hall squeezed his shot underneath Josh McNamara in the closing stages to deny us the three points.
It was an intense opening in Kingston-Upon-Thames, with Chelsea's youngsters applying plenty of pressure to Brian Barry-Murphy's side, pressing high up the pitch with physical intensity.
But City showed plenty of composure, keeping calm under the threat posed by the hosts, playing out from the back and finding routes up field.
Carlos Borges looked particularly dangerous, sprinting through on goal from the left wing before firing over Marcus Bettinelli's goal.
City went close again when Borges laid it off for Will Dickson to shoot, but Bettinelli reacted well to save the striker's effort, palming the ball into Dire Mebude's path. The winger pounced to poke the ball into the back of the net but the offside flag was already raised to deny the Blues an opener.
Another great chance fell City's way just before the half-hour mark with Chelsea giving the ball away inside their own box. Mebude squared to Bobb, but his left-footed effort was denied by a great one-handed save from Bettinelli.
City were made to pay for those missed chances as just after the half-hour mark, as Chelsea went ahead.
Zak Sturge's shot was blocked by Kane Taylor but Vale was on hand to slot the ball home to give the home side the lead.
It was a punishing blow for the EDS, who had been on top for most of the game, but City drew level just before the break in spectacular style.
Breckin broke forward from just inside the Chelsea half before releasing a stunning effort from outside the box which flew into the top corner beyond Bettinelli, ensuring the game was all square heading into half-time.
Into the second-half, City went ahead when a set-piece corner routine found Bobb, who worked himself some space inside the Chelsea box before lashing a fierce strike past Bettinelli and into the roof of the net.
City kept attacking in search of a third with Mebude denied by a fine save from Bettinelli before the Chelsea stopper once again denied the pacy winger from close range a few minutes later.
The Blues looked likely to hang on for a deserved win, but with ten minutes left Hall picked up the ball from outside the area before firing a left-footed effort at goal, squeezing underneath McNamara to draw the hosts back level.
City almost provided the perfect response when a nice link-up with Mebude played through Josh Adam, but the substitute dragged his shot wide of the far post, ensuring the points were shared.
HOW WE LINED UP
McNamara made his first EDS start of the season in goal, with a back four comprising of Juan Larios, Finley Burns, Taylor and Tom Galvez providing the protection.
It was a familiar midfield with Shea Charles sitting deep behind Breckin and the more advanced Bobb.
Mebude and Borges were named on the wings, supporting striker Dickson, who made his first start for Barry-Murphy's side this season.
Starting XI: McNamara, Larios, Burns, Taylor, Galvez, Charles (C), Mebude, Breckin, Dickson, Bobb, Borges.
Substitutes: Alleyne, van Sas, Adam, Smith, Agyemang.
Starting XI: Bettinelli, Rankine, Gilchrist, Mbuyamba, Humphreys (C), Webster, Hutchinson, Chukwuemeka, Burstow, Vale, Sturge.
Substitutes: Elliot, Beach, Wareham, Hall, Castledine.
WHAT IT MEANS
City's unbeaten start to the Premier League 2 season continues, with eight points sealed from the first four games.
The Blues currently sit fourth in the standings, just two points off the current leaders Crystal Palace.
The next test awaiting Brian Barry-Murphy's side is a Papa John's Trophy tie where the EDS will travel to face Mansfield Town.
Kick-off at the One Call Stadium will be at 19:00 (UK) on Tuesday 30 August.
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