Tom Ince put Derby ahead with his seventh goal in as many games when Johnny Russell's through-ball gave him time to shoot low past David Button.
The hosts grabbed a deserved equaliser when Ryan Fredericks teed up Floyd Ayite for a tap-in.
Stefan Johansen's shot from the edge of the box put Fulham in front, but Alex Pearce headed in Ince's corner.
Fulham, who are unbeaten at home in six matches, started at a furious pace but Ince struck for the Rams with their first noteable chance.
Scott Carson then saved from Fredericks and Kevin McDonald, but the Derby keeper was finally beaten for the first time in six matches by Ayite.
The Cottagers again forced the pace after half-time, with Scott Malone having a shot well blocked before Johansen's strike threatened to inflict only a second defeat on Derby boss Steve McClaren since his return.
But Pearce's equaliser forced a fifth draw in the last six matches between the two sides at Craven Cottage, despite a late scare for the visitors when Lucas Piazon's free-kick hit the bar.
The visitors, who have lost just once in 13 matches, drop two places to seventh, while Fulham, with just two defeats in 11, move up a place to ninth.
Derby manager Steve McClaren:
"What the boys have done for the past seven games has been incredible. To win seven in any division is difficult - to win seven and keep six clean sheets is even harder.
"It had to end sometime and I'm pleased it didn't end in defeat. I'm so proud of the players and the staff.
"Fulham kept coming at us and had our backs to the wall. This was the first time during our run we'd been behind and the players reacted well and got their reward."
Fulham manager Slavisa Jokanovic:
"I cannot be completely satisfied. We paid a heavy price for making a mistake. We played all the football, dominated the game and created chances.
"But it wasn't enough to win the game and this isn't the first time it's happened. We must work to improve ourselves and make ourselves more solid and competitive.
"They scored twice after our mistakes and in this competition that's not acceptable."