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.
The 24-year-old Kane headed in Christian Eriksen's free kick in the 22nd minute to overtake Shearer's 36 goals in 1995 with Blackburn Rovers.
Spurs won for the fourth time in five league games and remaining one point behind fourth-place Liverpool, which routed Swansea 5-0. It also capped another late surge to personal glory after he scored seven goals in the final two games of last season to beat Romelu Lukaku to the Golden Boot.
It was Kane's eighth treble in 2017 and so insatiable is his hunger for goals that it's sometimes easier to simply reel off statistics.
"First of all I want to congratulate Harry Kane, a massive achievement for him, well deserved", the Argentine coach said after the game. He almost grabbed a hat-trick midway through the second half, volleying Rose's cross an inch wide, before clinching it with a cool left-footed finish from Alli's pass.
Dele Alli and Son Heung-min each struck to make it 4-0 to Spurs, while Saints pulled goals back through Sofiane Boufal and substitute Dusan Tadic. When he topped last season's goal standings with 29 in 30 games, his scoring rate of 0.97 was the highest achieved by any Golden Boot victor in the league's 25-year history. After a stuttering campaign so far, the recent performances against Burnley and Southampton in the space of three days suggest that Spurs have once again found an extra gear at the end of the year.
"They are fantastic players here".
In this kind of form, it would take a fearless man to bet against the Spurs forward clinching the Premier League Golden Boot come May.
It only seemed to be going one way for Pelligrino's men as they came out for the second period, but they were inches away from finding an avenue back into the contest as Lemina curled a great effort from outside the area but was denied by the crossbar. Saints did the opposite, backing off Son, Alli and Eriksen naively and allowing Spurs to pick them off at will.