The England striker - a two-time Premier League Golden Boot winner - had never netted in the opening month of the season, but ended that hoodoo with the final goal 13 minutes from time - a calm side-footed finish inside the right-hand post.
Fulham had looked on course to claim a first point of the campaign after Aleksandar Mitrovic cancelled out Lucas Moura's fine opener, only for Spurs to triumph courtesy of a superb Kieran Trippier free-kick and Kane's intervention.
"It feels nice to score. It is a great win for us," Kane told BBC Sport as Mauricio Pochettino's side made it two wins from two in 2018-19.
"I had a few chances and I would have liked to have done better, but I was there to score the third.
"I can put it in the back of my mind now. A lot of people talked about it. It always seemed to happen in August for me. I could have probably had a hat-trick today on another day."
Kane added to Sky Sports: "I thought we played really well and we created a lot of chances. But credit to Fulham and only after about 60 minutes did we start to take it into more dangerous areas.
"It looked like it would be one of those days for me, but that is the game of a striker.
"We have been playing with each other for a long time, this team. The gaffer has said he is not going to just bring in players for the sake of it. It is a good start with two wins out of two."
Fulham boss Slavisa Jokanovic, meanwhile, took heart from his team's performance, but conceded that Spurs' greater class ultimately proved decisive.
"In the important moments Spurs showed great quality. After scoring a quality second goal, they scored a quality third goal," he told BBC Sport.
"We had good moments in the game, especially at the start of the second half. We scored and had chances to score a second.
"We always try to find a way to win the games. We started the second half well and scored, but to be honest, they changed shape to a diamond and we did not have enough power to stop them.
"My team showed great effort and pushed hard, but it is not easy for a new Premier League team, but this must encourage us."