The Spurs striker slotted in his side’s second goal in a four-minute spell to confirm their victory.
Lucas Moura put Spurs ahead with a curling 43rd-minute shot for his first Premier League goal.
A grounded Aleksandar Mitrovic nodded in a deserved 52nd-minute equaliser for Fulham, after Ryan Sessegnon's effort from a Joe Bryan cross fell into his path.
But Kieran Trippier curled in a superb 74th-minute free-kick to restore Spurs’ lead and Kane struck to confirm their second win of the season.
Fulham are pointless from two matches.
Spurs: 27 Aug v Man Utd (A)
Fulham: 26 Aug v Burnley (H)