Tony Weston opened the scoring for the Rams after 28 minutes played, before Zach Baumann levelled the game with a low drive that sailed into the bottom corner of Jack Thompson's goal.
County then scored three times in just seven first-half minutes, with Weston, Cruz Allen and Adebayo Fapetu all finding the net.
Ethan Williams's deflected strike with just 20 minutes to play reduced the deficit, but it was the visitors who claimed all three points.
Bidding to build on our recent victory against Manchester City last time out, Travis Binnion's side were asked questions of early on, with Sonny Aljofree vitally intercepting on two occasions to keep Derby at bay.