Robinson, who has been made available for transfer by the Rams, headed home Michael Jacobs' free-kick in the 10th minute and was on hand again from close range just four minutes later to tap in Nathan Tyson's low cross.
Former Chesterfield defender Derek Niven replied for the Evo-Stik League Premier Division side just before the half hour mark. He got on the end of a corner eight yards out but it has to be said that it was not good defending by Derby.
Shortly afterwards Robinson turned provider when his cross was converted by Jacobs.
Buxton were denied another goal when substitute Mark Reed saw his penalty saved by Saul Deeney before Robinson completed his hat-trick with 10 minutes remaining, capitalising on a poor back pass and sliding his shot beyond Andy Warrington.