Borough found themselves two down very early on and fought back but could find any more than the one goal, scored by Pemi Aderoju in the second half.
The defeat leaves one off the bottom of the table and now five points off the safety zone.
Tommy Widdrington’s side head to Dover on Saturday for their final game before Christmas – which brings a home Sussex derby with Worthing on Boxing Day.
See pictures by Lydia and Nick Redman on this page and those linked and get the Borough latest in the Eastbourne Herald on Friday.
17. Lydia Redman
Eastbourne Borough v Chelmsford City, National League South Photo: Lydia Redman : Lydia and Nick Redman
18. Lydia Redman
Eastbourne Borough v Chelmsford City, National League South Photo: Lydia Redman : Lydia and Nick Redman
19. Eastbourne Borough v Chelmsford City pictures by Lydia and Nick Redman (1).jpg
Eastbourne Borough v Chelmsford City, National League South Photo: Lydia Redman : Lydia and Nick Redman
20. Lydia Redman
Eastbourne Borough v Chelmsford City, National League South Photo: Lydia Redman : Lydia and Nick Redman