Ghanaian winger Enoch Andoh was on target for Paul Davis's men against Long Eaton United at the Arden Garages Stadium.
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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'.
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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.