Kane, who burst into the spotlight last season with 21 Premier League goals, had managed only one before the trip to Bournemouth, but after converting a first-half penalty he added two more from close range in the second half.
"We never had any doubts about him," Pochettino said. "It was a very important hat-trick for him, it's important for him to learn about that period, the good and the bad about football.
"I'm happy because he has become a much better player. This period was very tough for him."
Kane's performances have been decent this season without finding the net, but he said luck had finally turned his way on Sunday as he helped his side up to sixth in the table.
"As a striker you are going to go in and out of spells. I've not had a bit of luck that maybe I had last season and today was the day it changed with chances coming my way," he said.
Bournemouth, playing their first season in the top-flight, took the lead in the first minute through Matt Ritchie, but Kane equalised after being brought down by Artur Boruc.
Mousa Dembele and Erik Lamela had Spurs 3-1 ahead by half-time and Kane completed his hat-trick with two more goals in seven minutes.
Tottenham are unbeaten in the Premier League since the opening day of the season, and Pochettino said he was pleased with how they bounced back from Thursday's poor display in a Europa League loss at Anderlecht.
"It was the opposite of Anderlecht but I was very pleased with the reaction of the players, which was fantastic. We fully deserved the win and created even more chances," he said.
"It was a very difficult game against a team that looks to play on the grass, runs a lot, is very aggressive and our effort was fantastic in the way we pressed."
(Reporting by Steve Tongue; additional reporting by Martyn Herman; editing by Clare Lovell/Ian Ransom)
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