Taiwo Awoniyi gave Forest the perfect start when he headed them in front after 11 minutes from a Ryan Yates corner.
But Fulham struck three times in six minutes early in the second half to turn the match on its head.
Tosin Adarabioyo headed the equaliser in the 54th minute from a Willian cross and three minutes later Joao Palhinha found the top corner with a superb left-footed effort.
Forest were at sixes and sevens and Fulham scored again when Harrison Reed fired home from a Bobby De Cordova-Reid cross.
Lewis O'Brien gave Forest hope with a 77th-minute effort but Steve Cooper's side could not salvage a point.
Forest, who lost to Bournemouth in their previous game from 2-0 up, are second-bottom of the table with four points.
Fulham, who came up from the Championship with Forest last season, moved up to sixth with 11 points.