The striker tapped home the equaliser before netting an 86th-minute winner with a superb header as Spurs fought back from conceding an early opener to Mesut Ozil.
Kane, who has now scored 22 goals in all competitions this season, said on BT Sport 1: “That feeling is one I definitely won’t forget for the rest of my career.
“My first north London derby, to win the way we did, it’s incredible. I probably won’t sleep tonight.”
Kane’s rise this season has been meteoric, but the 21-year-old insists he is having no problem keeping his feet on the ground.
“I enjoy my football,” he said. “Just play with a smile on your face, do the best you can, work as hard as you can for the team. It’s been working well for us as a team, an incredible bunch of lads, the spirit’s high.
“I just want to be out on that pitch, playing with my mates. We are playing very well, we just want the next game to come.”
Kane’s team-mate Ryan Mason hailed the Tottenham fans, saying: “I don’t think there will be many better atmospheres around the world than what it was today.”
Tottenham boss Mauricio Pochettino described the win as a “special moment” before backing Kane to get better and better.
“His quality and his potential is massive, he can improve more,” he said on BT Sport 1.