Antonio Conte's side could have moved above Manchester United and up to second but were outplayed by a Foxes team that wasted several good chances and ended the game with 10 men after Ben Chilwell picked up a second booking.
The visitors responded with their first sight of goal just a minute later when Tosun steered the ball into the path of Wayne Rooney, but the former England captain dragged his subsequent effort wide of the target. "His life is probably fantastic at the moment".
The latest record comes after a stellar year in 2017 for Kane.
Son Heung-min was the best player on the pitch as he scored the opening goal, set up Kane's first and hit the post.
Speaking ahead of the game, the 63-year-old Everton boss said: "Tottenham look like they have overcome the problems they had in the early stages of playing there (Wembley)".
And soon after it Everton were undone when a superb turn from Son on the left enabled him to run into the area and cross low for Kane - who appeared to be a fraction offside - to steer home a low finish.
Fifth-placed Tottenham moved to 44 points and are level with Liverpool who take on league leaders Manchester City on Sunday.