"It's been a fantastic nine or 10 years in the Premier League. Hopefully many more years to go. I always love scoring," Kane said when asked about moving past Aguero's mark.
Wolves had frustrated the hosts for the majority of the afternoon but, similar to last week's derby, Spurs found a way and went home with the points.
Antonio Conte's men remain unbeaten in their opening three Premier League fixtures while Wolves have failed to earn a win so far this season.
There were very few clear-cut chances in the first half with headers from Kane and Ruben Neves the closest both teams came to scoring.
Spurs' players were booed off at half-time but responded well in the second half as Kane hit the crossbar with a close-range header.
Son Heung-Min came close to scoring after 63 minutes but his shot struck the outside of the post from a tight angle.
A minute later, Spurs finally took the lead when Kane got on the end of Ivan Perisic's flicked header from a corner to nod into an open net. It was also Tottenham's 1,000th Premier League goal at home.
New signing Richarlison came off the bench late on and almost grabbed his first goal for the club but missed the target.
Spurs withheld late pressure from the visitors to hold on and claim their second win out of three league games this season.
"Credit to Wolves, they made it difficult. In the second half we came out with intensity," Kane added. "We created chances and pressed better. We deserved the goal.
"Credit to the boys, they dug deep. We need to start games better. We need to improve. But the sign of a good team is winning when you don't play at your best."