Ashmore had a fairly straightforward game on his debut for the Kent club as he successfully shut out his former colleagues, while a penalty from Harry Kedwell and a second from Jean-Paul Kissock in the first half was enough to send Gosport home empty handed.
Alex Pike handed a Borough debut to keeper George Legge, who has joined the club on loan until April 2 from Readings.
In the same division, the Hawks’ recent improvement ended in a disappointing 2-1 defeat against Hemel Hempstead at Westleigh Park.
The visitors took the lead with a 15th-minute set-piece goal and the Hawks had Shamir Mullings sent off 10 minutes from time.
In a dramatic finish, Josh Hill equalised, only for the away side to snatch the win seven minutes into stoppage-time
Elsewhere, Howard Neighbour scored twice for Petersfield as they secured their third league home win in a row with a battling display against Bedford Town.
In the Ryman League premier division, Bognor were held to a 1-1 draw at VCD.
In the Wessex League premier division, AFC Portchester lost 3-0 at Sholing, Horndean won 2-0 at Whitchurch and Fareham drew 2-2 at home to Brockenhurst.