Reports Dan Deakins...
In a prolific first half for City, two close range strikes from Scott Fenwick and a fluky shot Anthony Church had the home side on course for their first home league win in 71 days.
The Sports were the only unbeaten away team in the division heading into this game. Dean Cox grabbed a lifeline for his side before half time from the penalty spot and Ryan Hall finished off from a few yards out to set up a very nervy last few minutes.
A strong, confident start was exactly what City needed in the circumstances, and within three minutes they had the lead through Fenwick. Chez Isaac got in behind the Eastbourne back line and squared for the striker to tap in from fewer than five yards.
Michael Spillane went inches from doubling the lead inside eight minutes. As a corner was swung in the midfielder dived to glance a header just past the far post.
Rhys Murphy too had a gilt-edged chance go wide on 10 minutes, driving at Mark Smith's goal and firing into the side netting.
Fenwick completed the triad of clear goal-scoring opportunities in as many minutes when his fierce right-footed effort struck the bottom of the near post. City were playing with exciting intent and had been unlucky not to be 3-0 up before the quarter hour mark.
Hall saw Eastbourne's first chance on goal with 18 minutes gone. In a classic counter-attacking move the Sports worked the ball to the edge of City's box and crossed for Hall, who's shot was deflected wide.
Church doubled the lead after 27 minutes. An Eastbourne defender inadvertently cleared the ball for a corner, from which the ball was initially cleared to the edge of the box. It landed at the feet of Church though, and the captain's looping shot went over everyone into the top corner.
Fenwick got his second exactly 10 minutes later. Murphy won the battle against the defender on the right wing, and sent a curling cross into the box for his strike partner who nodded past Smith.
Eastbourne needed to start clawing some goals back if they were to maintain their unbeaten away record.
Cox started the comeback from the penalty spot just a couple of moments after Fenwick had scored City's third, when Chris Whelpdale was adjudged to have fouled Charlie Walker in the box.
City had their own close range set-piece opportunity barely a minute later when Kiran Khinda-John was fouled right on the edge of the area. After much gamesmanship by the Eastbourne players in the wall - shuffling forward, being told by referee, Philip Staynings, to move back, and then shuffling forward again - Murphy's ferocious drive was tipped around the post by Smith.
As the sun glared just above the tree line at half time, it was as if it was setting on the Clarets' poor home form.
The second half soon settled into the scrappy affair many may have assume it would have been from the start.
The match hit the hour mark and it seemed as though City were happy to sit on their 3-1 lead. Their play was safe and focused on smart defending.
In a rare second half chance for the Sports, Kane Wills opened room for Yemi Odubade on the edge of the box, who forced Nathan McDonald into a save, albeit a routine one.
Fenwick could have had his hattrick in the 85th minute. From a throw in close to the corner flag, Fenwick was put in down the left flank and flashed a cross-come-shot across the goal. Whichever it was intended as, it very nearly settled the game.
Hall made sure the last four minutes of normal time would not be a simple case of seeing the result out, as the striker pulled the score line back to 3-2. Wills it was again creating the opportunity at the far post and Hall bundled in a scrappy shot from close range.
Eastbourne were now encamped in City's half, piling on the pressure and desperately trying to force an equaliser.
Into added time and Eastbourne had a throw on the left, parallel with City's box. It was launched in and after an almighty melee, Chelmsford scrambled it away.
All the late pressure was for nought though, and City withheld the onslaught