With national boss Roy Hodgson watching on from the stands at White Hart Lane, Kane moved on to 19 top-flight goals this season - making him the division's top scorer.
"Fantastic performance. I am very pleased for him [Kane] - first time he's had three goals in the Premier League [in one game]," Pochettino said.
"Harry is part of Tottenham, I'm happy for him but I think we worked hard. It was a difficult game, but it's true Harry is a very important player for us.
"He's in a very good moment in his career."
Tottenham had squandered a two-goal lead to allow Leicester to draw level before a penalty handed Kane his hat-trick goal, with Jeffrey Schlupp's own goal ensuring David Nugent's injury-time strike was only a consolation for the league's bottom side.
"Maybe it's part of the process because we are a very young team and we need to improve," he added. "After this season we need to improve our defensive game. After Manchester United it was important to win."
"We don't have news, he's in the hospital and we wait for the doctor now," he said. "We hope that nothing big, but we wait for the news. It's a big cut on the tendon on his knee."
