Defeats to Arsenal and then Manchester City, with two goals scored after 45 minutes at the Etihad, put more pressure on Conte and raised more questions about his future at the club.
Reports in Italy suggest he is likely to leave at the end of the season, but Spurs held on for a morale-boosting 1-0 win at Craven Cottage as Kane’s first-half strike was enough to win.
This moves Tottenham to within three points of Newcastle and Manchester United in the top four, although they have played more games than those two sides.
Conte punched the air as he celebrated passionately in front of the away fans after the game, his reaction largely relieved after his side put a terrible week behind them.
The Spurs boss has spoken of how the deaths of Gian Piero Ventrone, Sinisa Mihajlovic and Gianluca Vialli have put his life into perspective, and made him reflect on being away from his family in Italy.
Speaking after the game, Kane said it was up to the whole team to support their manager and find solutions to the current problems as a group.
“He’s going through a tough time in his personal life outside of football,” Kane told Sky Sports.
“Being part of a team, it’s not just the players on the pitch. It’s about being there for managers, employees and each other.
“We want him to win every game, he puts a lot of passion and effort in every game and every training session. We want him back with a win.
“We had a good meeting with the manager in the week between us, just to talk amongst ourselves to go back to what we’re good at.
“We’re going to keep fighting. The manager is a fighter, he is a really passionate manager. We should keep working hard for him. “