After a gritty 90 minutes with plenty of chances, Kane Richards' strike in the first half of extra time proved decisive.
It was harsh on Steve Bould's side, who had dominated the first half of normal time without capitalising on their opportunities.
Derby grew in the second period and those misses became costly as the Rams stole the tie in the added half-hour through lively striker Richards.
Bould was able to select a strong line-up for the occasion, with Reserves regular Martin Angha named among a powerful backline. Zak Ansah, Kyle Ebecilio, Jon Toral and Samir Bihmoutine have all been involved with Neil Banfield's second-string in recent weeks and all four started.
Derby, meanwhile, included first-team hotshot Mason Bennett in their starting line-up - although the 15-year-old is still at school.