Louie Theophanous gave the Clarets a second-half lead, poking home Chris Dickson’s knock down to ease the stress levels in the Melbourne Community Stadium.
Three minutes later substitute Shaun Jeffers doubled the lead with a vicious volley from range, before the striker added a City third in second half added time.
Neither side managed to successfully register any clear-cut chances, with Poole edging it marginally as the half-time whistle was blown.
The Clarets’ only real chance came towards the end of the 45 minutes, with Chris Dickson watching his header glance off the crossbar from a Robbie Willmott corner.
Poole should have arguably found themselves ahead when Marvin Brooks darted through on-goal. The experienced midfielder found himself one-on-one with Ross Fitzsimons following a defensive lapse in concentration, however Brooks failed to retain his composure by shooting wide of the post.
The home side came out of the starting blocks quickest in the second half, instantly searching for an opening goal. However, Ashley Miller was unable to get a shot on-target from Theophanous’ through-ball.
Rod Stringer’s side continued to search for a breakthrough and looked it most likely to come from a set piece. Willmott provided another chance from his fine corner deliveries, but Nick Hutchings was on hand to punch clear.
Abdulai Baggie caused brief trouble with his deep cross which Ross Fitzsimons could only palm away, however Luke Daley calmly put the ball out for a corner. However, from the consequential set-piece, substitute striker Richard Gillespie headed agonisingly wide for the visitors.
The home side’s determination eventually heaped the rewards though as a Willmott corner was headed down by Dickson to Theophanous, with the Greek striker poking home to send the Clarets faithful into jubilant scenes.
But City Boss Rod Stringer was still not happy with the solitary goal lead; introducing Shaun Jeffers for Chris Dickson with ten minutes left.
And the man in the Chelmsford hot-seat had his tactical change show rewards when Jeffers rifled home a stunning half-volley to all-but seal the result from Willmott’s long ball.
Fellow substitute Elliott Buchanan almost added a third for the Clarets, but he was unable to keep his shot low after Theophanous did well to drag the ball back.
Jeffers signed, sealed and delivered the three points in added time as he sent a finely executed finesse shot round Hutchings.