Hawks will travel to Grimsby Town in the second round of the FA Trophy after winning 2-1 at Braintree.
Christian Nanetti and Stefan Bailey netted to ensure Hawks upset their Blue Square Bet Premier hosts.
Fareham went down 3-1 to Bournemouth Poppies in the Wessex premier.
Liam Robinson had put the Reds ahead but Wayne Boud was sent off as they crumbled to defeat.
Hayling were 3-2 winners in a thriller against Romsey Town at The College Ground.
Alex Easton scored twice for Hayling and Nathan Kirby got their other goal.
Adam Wilde, Justin Bennett and Luke King all found the net as Gosport swept past Blackfield & Langley 3-0 in the Hampshire Senior Cup at Privett Park.