The Marketmen, boasting the league's best away record and unbeaten on their travels, looked on course to maintain that form when Keiran Leabon gave them a seventh-minute lead with a rasping 20-yard daisy-cutter.
But Jordan Cox equalised in the 27th minute, turning in Junior Dadson's low cross from close range.
A draw would have been a fair result but Millers nicked it at the death when Appiah broke through on the right and fired past Louis Johnson in the Needham goal (pictured by Mikey Cartwright).
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