Martin Jol's hosts began brightly and led through a stunning Steve Sidwell effort, only to slump in the second half and concede to Gabriel Paletta before Sansone settled it four minutes from time.
The game was just 10 minutes old when Sidwell struck - the midfielder surging to within 35 yards of goal before unleashing an unstoppable shot that left Antonio Mirante with no chance.
The Italians responded quickly and Amauri was only denied an equaliser by an offside flag before Marco Parolo rattled the crossbar and Aleandro Rosi failed to find the target with a header.
Recently-arrived Cottagers goalkeeper Maarten Stekelenburg then proved his worth with a smart stop from Jonathan Biabiany shortly as Parma continued their ascendancy before half-time.
Parity was eventually restored four minutes before the hour-mark - Italy striker Antonio Cassano providing the cross for captain Paletta to soar highest and head home.
Fulham struggled to regroup and were on the defensive for much of the remainder of the game - Sansone and former Fulham striker Stefano Okaka closest to firing the visitors ahead.
In the 86th minute, the winner came when Gianni Munari knocked down a Djamel Mesbah and Sansone pounced, and Pavol Bajza ensured victory with an excellent stop from Pajtim Kasami at the death.