The visitors started brightly as Parkes fizzed a volley just wide before Dickson over-ran his attack forward. The City striker came close moments later, but his shot curved wide of the left post, then Oliyide cut inside and shot straight at Jessup.
Dean Brennan’s side looked most likely to open the scoring as Chelmsford proved unconvincing in defence.
And the visitors eventually scored on 26 minutes. A deep right-wing cross met Moyo’s run and the Tudors’ front-man climbed highest in a crowded box to head home.
But four minutes later Chelmsford drew level as Dickson raced onto a loose ball and unleashed a fierce strike past Barnes.
City threatened as Miles’ freekick was not cleared but the visitors restored their lead when McCall’s long-range effort snuck in at the far post having appeared to have been heading wide.
Another four minutes later City levelled. Miles’ deep corner from the right wing found Church in a crowded area but the Chelmsford skipper headed home to level the scores again.
Oliyide should have put the Tudors in-front early in the second half but he was unable to connect to the bouncing cross, moments later he then curled a second chance wide as the visitors searched for another goal.
The Tudors’ third goal eventually came on 53 minutes as Jessup failed to punch Parkes’ freekick and it fell to Yakubu at the far-post who calmly headed into an empty net.
But the Clarets levelled with five minutes left on the clock. Substitute Green knocked the ball past his marker before striking a low shot into the bottom corner to rescue a point for City in their push for promotion.