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Roos was kept busy in the opening exchanges, as he was then called upon to deny Dembele again and then Stephen Arthurworrey.
In the 27th minute, the young forward pounced on Craig Forsyth's dreadful backpass and calmly finished to give the Cottagers the lead.
As half time drew closer, Johnny Russell was handed two chances to restore parity for Derby.
First, he bent his strike narrowly wide after neat work from Simon Dawkins before firing at Kiraly when one-on-one.
The miss proved costly as Dembele doubled the hosts' lead a minute before the break, but Derby were gifted a lifeline in stoppage time as the referee spotted a handball inside the area and Martin halved the deficit with the last kick of the first period.
Two minutes after the break, Russell levelled things up after a powerful run from Dawkins and the winger completed the turnaround a few moments later with a thumping finish from inside the area.
With Fulham still wobbling from Derby's quickfire blitz, Jeff Hendrick rounded off a slick move to put his side two goals ahead.
Russell and Dawkins combined once more shortly after, with the latter flicking in an attempt at the near post to put the shine on an incredible comeback by Steve McLaren's men.