Karleigh Osborne's side made the trip to Wiltshire on Sunday to take on a team two divisions higher than them in the footballing pyramid.
The Bees were rewarded for an enterprising start midway through the first-half when Rebekah Wiltshire broke in behind and finished calmly. That was the scoreline until the hour when Emily Boycott brought down Holly Baker in the area; Hayley Huckin netting the resulting spot-kick.
The Bees could have won it in normal time but a superb save from home keeper Abi Footner and a controversial disallowed goal meant the game went to penalties.
After both sides missed their opening efforts, the next 13 spot kicks were all scored. Trailing 7-6, home defender Mollie Smith sent her effort over the bar as The Bees progressed.
The Preliminary Round draw has thrown up a local derby with Brentford at home to Ashford Town (Middlesex) on Sunday 20 September.