The Rams, featuring eight players with first-team experience, had far too much for their Evo-Stik Premier Division opponents and romped to a 5-0 victory on their return to the competition after a 25-year absence.
Academy striker Callum Ball put Derby ahead after only 90 seconds, tapping home Lee Croft`s low cross.
Jeff Hendrick mopped up the rebound from a Ben Pringle effort to double the advantage after 15 minutes.
Conor Doyle extended the Rams` lead before the break following a mistake by Buxton goalkeeper Scott Hartley before Arron Cole volleyed home another Croft cross to make it 4-0 in the 73rd minute.
Hendrick saw his penalty saved before completing the scoring with two minutes remaining after Hartley pushed Pringle`s shot into his path.
Source: PA
Source: PA