Hot-shot Diego De Girolamo hit the winner five minutes from time as new boss Steve Cunningham marked his first game in charge with victory.
It was the first time the Bucks had made it this far in the competition since 1962.
Buxton, who controversially sacked Gary Hayward days before the match, started strongly with Jamie Ward having a shot blocked inside 25 seconds.
York keeper Pete Jameson made a good save to keep out De Girolama blocked on 11 minutes before Theo Richardson made a great save from a Clayton Donaldson header.
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Ward and Chris Dawson enjoyed further chances before the break as the visitors made light work of the difference in levels.
Richardson saved Sam Fielding’s close range header four minutes after half-time before Di Giralamo was off-target just after the hour mark.
But the striker sealed a historic win for the part-time battlers when he smashed home in the closing stages.