Ben Marlow and Sam Higgins scored against their former employers to propel the Rocks off the bottom of the Vanarama National League South.
Marlow opened the scoring inside the first minute as he capitalised on lethargic defending; Higgins doubled the hosts' lead by beating the offside trap to poke home from an angle.
City Head Coach Nick Haycock made one change to the side that defeated Gloucester City a week ago; Chez Isaac dropped to the bench as Scott Fenwick returned from suspension. Meanwhile, recent signing Tobi Sho-Silva took Mark Haines' spot on the bench and Michael Spillane joined the substitutes' at the expense of Luke O'Reilly.
A continual downpour of rain made for a difficult atmosphere, with the wet pitch adding extra pace to the speed of attacking play.
The Rocks entered the fixture having won just three of their home matches but started the game with blistering intent. A long throw-in from the right wing bounced once inside the Chelmsford box, allowing Marlow to drift to the far post and send a looping header over Ashley Maynard-Brewer to open the scoring on 53 seconds.
City came close moments later as Max Porter split the Rocks' defence, but Jack Giddens remained composed under pressure from Tom Wraight to chest the ball and clear the danger. The Clarets came close again when Steve Sheehan wrong-footed goalkeeper Giddens with a shanked clearance, but the blushes were spared as it went out for a corner.
Chelmsford continued to dominate but could not turn their domination of possession and chances into rewards. Jonny Giles struck well over as he latched onto Anthony Church's backward header, Matt Young then dragged a low shot along the ground and wide.
It could have been 2-0 to East Thurrock when Higgins was allowed freedom in the centre of the pitch, but Elliot Omozusi recovered well to produce a goal-saving tackle before Adrian Cascaval swept up the loose ball.
But Higgins made no mistake a moment later as he lapped up on City's poor defensive positioning to stroke home. The former City striker burst down the left wing after being played onside by Cascaval, Higgins cut inside as he reached the edge of the box and placed a shot beyond Maynard-Brewer into the far corner on 21 minutes.
City's number of chances vastly outnumbered the Rocks' but they continued to fail to break the deadlock as Giddens produced a number of good saves.
Scott Fenwick slid the ball forward to Wraight, but Giddens saved well from close range. Fenwick provided again as he delivered a low cross from the right wing, but Wraight and Giles could not stroke the ball home as another chance went begging. Lee Burns also arrived at the last moment to clear Fenwick's effort off the line after the Geordie striker poked the ball beyond Giddens.
Higgins came close to wrapping the fixture up in the dying embers of the first half with a thunderous effort from his own half. The experienced talisman controlled a long ball with his chest and swivelled on the spot before shooting; however, the striker was denied a certain Goal of the Season winner as the crossbar stood firm.
Nick Haycock looked to capitalise on his side's superiority in terms of chances, making a change at half-time as Amos Nasha made way to give Sho-Silva his debut appearance.
And the on-loan striker almost rewarded his coaching staff as he came close to pulling a goal back seconds after his introduction to the game. Church's initial effort was blocked for a corner but Sho-Silva rose highest in a crowded box to head the subsequent set-piece against the crossbar.
Giles' weaving run allowed the winger to find space to curl a shot towards goal, but his curling effort struck the bar.
Haycock made his second change on the hour-mark as Spillane came on for Max Porter.
Church was unable to get the first touch when he found space inside the box as a defender recovered well to concede a corner. But Sho-Silva should have grabbed a debut goal when he was left unmarked at the far-post, only to send a looping header over the bar.
In an incredibly frustrating fixture for City, Haycock replaced Fenwick with Chris Whelpdale as the Clarets went for a different attacking approach in order to try and claw themselves back into the tie
Sho-Silva had another close-range effort blocked inside the box while Church's volley left Giddens' palms stinging late in the second half, but the Clarets were left baffled as to how they had not scored in a 2-0 defeat to East Thurrock.