On his first league start for Tim Sherwood's side, the 20-year-old put Spurs into a 2-1 lead after they came from behind to cruise to the three points.
Kane has now featured 14 times in all competitions so far for Spurs this campaign after spending most of his time out on loan in previous years.
The youngster has gained experience at Leyton Orient, Millwall, Norwich and Leicester in recent seasons, but he is now aiming to become a regular fixture at Tottenham.
"Going on loan at 17 [to Leyton Orient], people didn't realise how young I was," Kane told the Tottenham Journal.
"People think I am a bit older but a first Premier League goal for Tottenham at 20 is not bad. Hopefully I can push on now.
"I've been waiting for the opportunity to get the start. It was great to get the goal and hopefully I can push on and get a few more.
"If you can do it in the Premier League you can do it pretty much anywhere. Hopefully I will get a few more chances and a few more goals."
Emmanuel Adebayor was also on the scoresheet against Sunderland and Kane says he enjoys playing alongside the Togo international.
"It's great playing with Ade, he's a world-class player and everyone can see that," he said.
"The manager has told me to be patient, because my time will come. I'm happy here, it's a great club and we're a top team.
"We've got a great crop of youngsters coming through and we have got to make the most of that."