The England skipper's goal against the Bluebirds meant that he has now scored at least once against all 28 of the Premier League opponents he has faced.
It was the perfect way for Mauricio Pochettino's side to bounce back after their disappointing 3-1 defeat to Wolves at Wembley on Satudray.
Kane needed just over two minutes to complete his set and Spurs completed the job with goals from Christian Eriksen and Son Heung-min.
The win lifted Spurs back up to second place and to within six points of leaders Liverpool.
Kane's goal was his 18th of the season and he started and finished the move.
Spurs extended their lead when Kane was involved again - creating a move which featured Moussa Sissoko and Son and ended with Eriksen firing the ball home.
It was the 100th Spurs goal he had been involved in – 45 goals and 55 assists. The third came after 26 minutes when Son finished his eighth goal in nine games.
Tottenham will be able to watch Thursday's much anticipated showdown between Manchester City and Liverpool with the pressure off, knowing a defeat for City will mean they are still in the title race despite their slip-up against Wolves.
And for Kane it starts the New Year in much the same way as he finished the last. He capped a memorable 2018 which included leading England to the semi-finals of the World Cup by being named in the Queen's New Year Honours List.