Recent defeats to Arsenal and Manchester City had further soured the mood at Spurs, and defeat at Craven Cottage would have seen Fulham move above them and into fifth in the Premier League table.
But that was not to be the case, Kane firing home from just outside the box in first-half stoppage time to draw level with Jimmy Greaves on 266 goals for the club, and that proved to be enough.
Fulham were the better side for much of the first-half, Harrison Reed and Willian among those to test the out-of-form Hugo Lloris before Joao Palhinha headed over a big chance from a free-kick.
Kane provided a lesson in clinical finishing just before the break, as he drifted inside onto his right foot and let fly from 20 yards out, curling an effort into the bottom corner.
Spurs were able to keep Fulham at arm’s length for much of the second-half, and went close to doubling their advantage. Kane was denied by a brilliant save from Bernd Leno, before the striker fired wide from an Eric Dier knock-down.
A late Fulham charge did not really materialise, as Tottenham put a disastrous week behind them with a victory that moves them to within three points of the top four.