They took a fifth minute lead when a good move was finished off with a tremendous shot from outside the box by Oliver Haines.
Dons came close to making it two, but Patrick Roberts and Daniel Haines were unable to convert chances created by Callum Wilson, and Hamsey took advantage by getting an equaliser.
Dons began to take control in the second half, with Jordan Gallager causing plenty of problems for Hamsey, whose keeper had to make four great saves.
But their persistence was rewarded with a fine winner from Billy Frost.
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