The hosts appeared to be cruising after Moussa Dembele’s first Fulham goals fired them into a 2-0 lead shortly before half-time.
But Chris Martin pulled one back from the penalty before the interval and the Cottagers capitulated after the break, as goals by Johnny Russell, Jeff Hendrick and two from Simon Dawkins sent the Sky Bet Championship leaders storming into the quarter-finals.
It was the second time Derby have put five past Fulham this season, although the Londoners are a completely different side to the one sent out by the hapless Felix Magath that day in August.
Revitalised under caretaker boss Kit Symons, they had won five of their last eight games, and despite eight changes to their line-up they roared into the lead thanks to teenage striker Dembele, as well as some shambolic Derby defending.
In the 27th minute Craig Forsyth made the mistake of not looking when he played an attempted backpass to Kelle Roos, Dembele intercepted the ball and, having rounded the Rams keeper, rolled it into an empty net.
If that piece of defending was poor, what followed in the 45th minute was downright comical. Roos inexplicably passed the ball to Omar Mascarell, the on-loan Real Madrid midfielder, who was promptly robbed on the edge of the box by the lurking Bryan Ruiz.
The stand-in Fulham skipper nudged the ball to Dembele, who had another simple task of slotting it past Roos to double the lead.
However, the spirit of generosity quickly spread to the Fulham defence when moments later, in first-half stoppage time, Tim Hoogland needlessly handled Mascarell’s free-kick and Martin stepped up to dispatch the penalty.
It proved a pivotal moment and fired-up Derby drew level within two minutes of the restart when Dawkins reached the byline and cut the ball back for Russell to rifle home from close range.
They took the lead in the 54th minute, Dawkins crashing in a superb angled drive from 20 yards which Gabor Kiraly got a hand to but could not prevent hitting the top corner.
Six minutes later the Rams went further ahead, when Forsyth flicked on Bryson’s cross and left Hendrick with a simple finish from six yards out.
And at the end of a whirlwind 20-minute spell the fifth arrived when Russell crossed from the right and Dawkins found the net with an extravagant backheel flick to put the seal on a stunning comeback.