It was his 266th goal in all competitions for his boyhood club, equalling the record total of Jimmy Greaves.
But it could have been an even better night for the Spurs star, who was denied a goal that would have made him only the third player to join the Premier League's 200 Club, after Alan Shearer and Wayne Rooney.
He was denied that honour by a sublime close-range save by Leno in the second half.
It means Kane will have to wait until Spurs' next league match, against Manchester City on 5 February, for another chance to reach that landmark.
Fulham would have gone above their visitors with a win but could not find a way past a Spurs side who have been leaking goals of late.
It takes fifth-placed Spurs on to 36 points, three behind Manchester United, who have played one match fewer.
Just a third home loss of the campaign leaves Fulham in seventh, with 31 points.