Carey kept out four out of five Tanners kicks while Borough put away two of their first four to clinch the victory.
They’d looked to heading through the first round tie eearlier whne they went 2-0 up in the 90 minutes thanks to Max Sheaf and Nathan Laleye’s first goals for the club. But the hosts hit back with two goals in the last 20 minutes to take it to penalties.
It takes Steve King’s side into Monday’s draw, with Burgess Hill Town, Chichester City and Whitehawk the other Sussex sides to come successfully through the weekend’s first round.
See pictures from Leatherhead v Eastbourne Borough by Lydia and Nick Redman on this page and those linked, and get all the local football action in the Eastbourne Herald on Friday.
33. Lydia Redman
Leatherhead v Eastbourne Borough, FA Trophy first round Photo: Lydia Redman : Lydia and Nick Redman
34. Lydia Redman
Leatherhead v Eastbourne Borough, FA Trophy first round Photo: Lydia Redman : Lydia and Nick Redman
35. Leatherhead v Eastbourne Borough pictures by Lydia and Nick Redman (6).jpg
Leatherhead v Eastbourne Borough, FA Trophy first round Photo: Lydia and Nick Redman
36. Lydia Redman
Leatherhead v Eastbourne Borough, FA Trophy first round Photo: Lydia Redman : Lydia and Nick Redman