Ghanaian winger Enoch Andoh was on target for Paul Davis's men against Long Eaton United at the Arden Garages Stadium.
Captain Will Grocott was also on target while Lewis Wilson scored the winner seven minutes from time.
The game ended 3-2 and the Bards booked their place in the fourth qualifying round.
The Bards will visit the National League North outfit on Saturday, 16th October knowing a win would see them progress to the first round 'proper'.
The winners will receive £9,375 and the losers take home £3,125.
So far Stratford Town have picked up £11,250 in prize money from this year’s history-making FA Cup run.