Kane, who squandered a golden opportunity in front of goal in the 51st minute, finally broke the deadlock just six minutes later with a powerful strike from outside the box.
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The England captain, who had not scored in his previous two Bundesliga games, the loss to Aston Villa in the Champions League or the recent two England matches earlier in October, then drilled in following a goal mouth scramble on the hour.
He secured his hat-trick in the 80th minute with his eighth league goal of the season after Joao Palhinha's shot was blocked with substitute Kingsley Coman curling in Bayern's fourth goal in the 89th.
"To be honest, I didn't waste any second thinking about whether he scored or not (in past games)," Bayern coach Vincent Kompany told a press conference. "I know what qualities he has. He's always hungry, even in training. Sometimes we forget how hard Harry works for the team. He has always scored.
"I am happy with how the game went. I was not surprised that it was not easy and at halftime the main thing for us was to stay calm."
Bayern had midfielder Aleksandar Pavlovic taken off after eight minutes with a broken collarbone that will sideline the Germany international for several weeks according to the club.
But they still had control of the game but could not make their dominance count in the first half.
Stuttgart, last season's runners-up, gradually ran out of steam and could no longer keep up after the break when Bayern struck three times in 23 minutes with Kane, now on eight goals in the league, to kill off the game.
The visitors' best chance was in the 53rd minute when Josha Vagnoman missed the target, firing over the bar from 10 metres out.
The Bavarians are top of the standings on 17 points, ahead on goal difference of RB Leipzig, winners 2-0 at Mainz 05.
Freiburg are third on 15 with champions Bayer Leverkusen in fourth a point behind after their 2-1 win over Eintracht Frankfurt.
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