While Tottenham were frustrated for long periods at Selhurst Park, Christian Eriksen's goal saw them respond well to the disappointment of losing to Chelsea in the FA Cup semi-finals on Saturday.
And now Kane and Spurs are keen to pinch the league title from under their London rival's nose, although the England international is determined that his side do not pay too much attention to Chelsea's fortunes.
"[The win] is unbelievable," he told Sky Sports. "This is a tough place to come and we had to keep fighting. We knew our time would come and what a strike by Christian Eriksen.
"We knew we had to bounce back after the weekend and what a performance by everyone. We knew we had to dig deep and get a goal. It was a perfect win in the end.
"We can't focus on Chelsea. We have five tough games, we have to win them to have a chance and hopefully we can do that."
Eriksen conceded Tottenham were far from their best at Palace, but he was delighted that his 78th-minute strike proved to be the winner.
"We struggled with some parts of our game - we didn't have many chances, but luckily we scored with the one we had," he said.
"We played more attacking in the second half which gave us more chances and I think we did very well."