Billingham Synthonia came from behind at Westfields in Herefordshire on Sunday afternoon to become the fifth STL Northern League team in the last 32 of the FA Carlsberg Vase.
It was a fairly even game in the first half, and Westfields took the lead just before half time with a header by Glover.
But Synners levelled three minutes later, when David Yale crossed for James Magowan to head past the Westfields keeper from six yards.
Synners were on top in the second half, and nearly won it when Colin Iley broke through and chipped the keeper, but the ball dropped just wide.
However, two minutes into extra time Synners grabbed a morale-boosting winner when a corner was cleared to the edge of the box, from where Nathan Jameson found the net with a right foot shot.
Synners assistant manager Lee Tucker said: "It was a very strong performance by us, and our fitness was better than theirs in the second half and in extra time."
Synners are now at home to Tadcaster Albion in the fourth round a week on Saturday.
Dunston could make it six STL teams through to the last 32 if they beat Heather St Johns in their re-arranged tie this Saturday
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