Mauricio Pochettino's side were not at their best for long periods of the game at Selhurst Park, with the Eagles staying compact and organised.
The north Londoners appeared to be drifting towards a costly draw after struggling to find their rhythm in attack.
But Christian Eriksen produced a moment of magic courtesy of a long-range strike to keep Spurs' title hopes alive.
Chelsea's lead at the top of the Premier League table stands at just four points and Kane was delighted Spurs managed to pass such a difficult test in south London.
"It's unbelievable. It's a tough place to come. We had to keep going and keep fighting," he told Sky Sports.
"We knew we had to bounce back after the weekend. We knew if we dug deep we'd get the goal. It was the perfect win."
Spurs face Arsenal in the north London derby this weekend while leaders Chelsea face a tricky trip to Everton.
But Kane is determined not to focus on Antonio Conte's side as the finish line approaches, with Spurs likely to need a flawless end to the campaign to pip their rivals at the post.
"It's not about sending out a message," the Spurs striker added.
"We can't focus too much on Chelsea - we've just got to try and win our last games."