The England international, who has made history in just 36 appearances compared to Shearer's 42 for Blackburn in 1995, put Spurs into a two-goal half-time lead with a close-range header and a simple tap-in before completing his second successive hat-trick with a dinked finish over Saints goalkeeper Fraser Forster for his side's fifth.
Kane's third goal also moved him ahead of Barcelona forward Lionel Messi for the most goals for club and country by any player in Europe's major leagues in 2017, delighting manager Pochettino.
"First of all I want to congratulate Harry Kane, massive achievement for him, well deserved," said the Argentinian coach. "And we are all so, so, so, so happy because it's an amazing thing to celebrate.
"For me, he's world class. I think today, (number) one striker in the world, specific position, who's better than him?"
Speaking to Sky Sports, Kane said it was a "great feeling" to break Shearer's record.
"It was hard not to think about it going into the game being level," he said. "As always I wanted to win the game first and foremost but of course I wanted to score as a striker.
"To get that goal early on and get that record was a great feeling and I could enjoy the rest of the game."