Jack Lancaster gave the Blues a second half lead before Rhys Murphy scored a late leveller for the Clarets.
With Ipswich Town boss Paul Hurst in attendance at the Melbourne Community Stadium, the young Tractor Boys were keen to impress against the City. The Blues named a team mixed of young and first-team players, with their recent summer recruit Jordan Roberts featuring for the Suffolk side.
Aaron Drinan came close to opening proceedings for the Blues after three minutes, but he dragged his low drilled effort wide of the left post. The Clarets had a chance of their own shortly after, however Jai Reason's quickly taken free-kick was directed into Harry Wright's hands from Tom Wraight's tame volley.
Reason came close to setting-up City's opener on fourteen minutes, but his pull-back to Amos Nasha was sent curling wide of the goal. Four minutes later Jordan Chiedozie had a great chance to score but his low shot was saved well by Wright.
The Blues came closest to scoring when Roberts' effort smacked the upright, although Ben Folami's follow-up effort was blocked at close range by Ross Worner.
City came close once again as Wraight found space on the left wing to pick out Jai Reason, although the midfielder's header looped over. Moments later Chelmsford wasted another chance, this time as Rhys Murphy cannoned an effort over the bar to send both sides into half-time level.
Chelmsford almost opened the scoring early in the second half when Anthony Church came agonisingly close to connecting with Wraight's cross, Wright then held onto Murphy's tame effort as City provided plenty of endeavour but without reward.
As the game approached the hour-mark, Nasha dragged a left-footed shot wide of the post before Murphy attempted an audacious lob of Wright.
And City were left frustrated as the visitors took the lead on 69 minutes. When Ross Worner over-stretched to control a back-pass, which left him grounded with an injury, Jack Lancaster stole the ball and tapped into an empty net.
With ten minutes left, Murphy rounded two Ipswich Town defenders before forcing Wright into a strong save - although substitute Taylor Miles was unable to convert the rebound from close-range.
A minute later, Murphy ferociously swept home Scott Fenwick's left-wing cross to salvage a late equaliser from a superb performance against Ipswich Town.