Kane's goal also marked his 100th home goal for Spurs and made him the leading scorer in the north London derby.
It was Son's 10th goal of the season -- as many as Arsenal have managed in the league. Eight of those have been assisted by Kane.
Kane perfectly controlled a Moussa Sissoko pass from a Spurs counter-attack before setting Son off down the left side of the pitch.
Instead of attempting a darting run behind the Arsenal defence, Son unleashed the ball 20 yards out and it went soaring past Bernd Leno in the Arsenal goal.
A minute into extra time at the end of the first-half, it was then Son's turn to provide an assist for Kane.
With Arsenal caught on the counter-attack again, Son held the ball until Kane appeared on his outside left and then slipped a pass to him. Kane then powered the ball into the roof of the net from close range.
"It is not just about scoring goals and being world class players. It is about being great guys and doing your best for the team," Tottenham manager Jose Mourinho said of his scorers.
"[Arsenal] gave us problems, problems we were able to resolve."
Partey had limped off the pitch during the second Spurs counter-attack, leaving Arsenal a man down when Kane scored.
Despite enjoying the majority of the possession in the second-half, they were unable to find a goal as Spurs were happy to play defence after their first-half display.
Victory means Jose Mourinho's side move two points clear of Chelsea who had claimed top spot on Saturday. They have 24 points from 11 games and are unbeaten in the league since losing on the opening day of the season.
Information from Reuters contributed to this report.