Fulham, relegated from the Premier League after 13 years in the top-flight, have lost all four league matches this season as pressure piles on German manager Felix Magath.
Jamie Ward opened the scoring for the hosts midway through the first half before Scott Parker levelled after halftime.
However two goals by Chris Martin and further strikes by Craig Bryson and Simon Dawkins secured Derby their second win of the campaign while Fulham join Blackpool as one of two teams without a point after they lost 1-0 to Wigan Athletic.
Michail Antonio scored twice for Forest, while Matty Fryatt and Britt Assombalonga also hit the back of the net as Stuart Pearce's side moved to 10 points from four matches.
Leeds United had Giuseppe Bellusci and Sam Byram sent off as they finished with nine men in a 4-1 thrashing by Watford, while Lewis Grabban scored the only goal as Norwich City beat Ipswich Town 1-0 in the first East Anglian derby for three years.
Wolverhampton Wanderers, promoted from the third tier, moved up to fourth after an injury time own goal from Cardiff City captain Mark Hudson gave them a 1-0 win, meaning they have beaten all three teams relegated from the Premier League.
It ensured a disappointing end to a miserable week for the Welsh side after the publication of offensive text messages sent between former manager Malky Mackay and sporting director Iain Moody before both left the club last season.