The Sports are now six points adrift of the safety line – and a win at home to fellow strugglers Havant and Waterlooville is a must.
It was goalless until seven minutes into the second half – and then the Clarets took control to leave the home faithful frustrated.
See pictures by Lydia and Nick Redman on this page and the ones linked and get the latest from Priory Lane in the Herald, out on Friday.
1. Eastbourne Borough v Chelmsford City pictures by Nick and Lydia Redman (12).jpg
Action from Eastbourne Borough's home loss to Chelmsford City in National League South Photo: Nick Redman:Lydia and Nick Redman
2. Eastbourne Borough v Chelmsford City pictures by Nick and Lydia Redman (14).jpg
Action from Eastbourne Borough's home loss to Chelmsford City in National League South Photo: Nick Redman:Lydia and Nick Redman
3. Eastbourne Borough v Chelmsford City pictures by Nick and Lydia Redman (16).jpg
Action from Eastbourne Borough's home loss to Chelmsford City in National League South Photo: Nick Redman:Lydia and Nick Redman
4. Eastbourne Borough v Chelmsford City pictures by Nick and Lydia Redman (35).jpg
Action from Eastbourne Borough's home loss to Chelmsford City in National League South Photo: Lydia Redman:Lydia and Nick Redman