Nicholas Sullivan's 15th minute strike gave the Anvils a half-time lead. Oli Leslie doubled Crawley Down Gatwick's advantage on 63 minutes before Daniel McCabe reduced the deficit three minutes later.
But Luke Donaldson's first goal for the Anvils, with 14 minutes to play, sealed the three points for the visitors.
Crawley Down Gatwick travel to Saltdean United in the second round of the Sussex Senior Cup tomorrow (Tuesday) evening before hosting British Airways in the FA Vase second round this Saturday.
Oakwood were beaten 2-0 away at Wick in SCFL Division One on Saturday. The defeat leaves the Oaks 14th in the table with 15 points from 16 games.
Oakwood visit fifth-placed Shoreham this Saturday.
Meanwhile, Forest Row, now in ninth, were held to a 1-1 draw at Selsey.
Ollie Monnery struck for Frow three minutes before half-time. Evan Harris levelled for the Blues on 77 minutes.
Forest Row host third-from-bottom Hailsham Town this Saturday.
Copthorne thumped bottom club Brighton Electricity 6-1 at home to move up to third in SCFL Division Two.
Stuart Cameron (2) and Daniel Sullivan put Copthorne three goals to the good before the break.
Sullivan and Oscar Weddell (2) netted in the second half for the hosts. Samuel Eseku hit Leccy's only goal of the game.
Copthorne are now eight points behind place-above Rustington, and a further point behind leaders Jarvis Brook.
Copthorne make the trip to St Francis Rangers this Saturday.
Charlwood exited the Surrey FA Intermediate Cup at the second round after a 1-0 loss at Frenches Athletic.
Charlwood visit Littlehampton United in the SCFL Division Two Challenge Cup this Saturday.