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Kane became the first English player to score in his first two Bundesliga games when he clipped a penalty down the middle just before half-time.
The ex-Tottenham forward wrapped up the 3-1 win after the break with a lovely finish with the outside of his foot.
Thomas Tuchel's side dominated possession throughout.
"Great game today," said Kane in a video on social media afterwards. "To score two goals in front of you fans was a special feeling.
"Lots of improvement still to come but to start the season with two wins is a really great start, so thanks for the support as always.
"You know [I] appreciate every single one of you there in the stadium singing my name, so look forward to next week now, recover well and let's keep it going."
After a sluggish start, Bayern got into their stride and opened the scoring when Leroy Sane's shot hit the post and went in off defender Felix Uduokhai.
Referee Sascha Stegemann then awarded a penalty for Niklas Dorsch's handball after being called to check his monitor by the Video Assistant Referee, with Kane converting the spot-kick with ease.
After the break Bayern looked far brighter, Sane curling a fine shot against the post, before Alphonso Davies found Kane onside and free in the middle to loft a calm finish into the net.
Substitute Dion Beljo wriggled free to poke in a late consolation for the visitors, while Kane wasted a chance to complete his hat-trick when he headed wide.
However, he has now scored in his last seven league appearances, stretching back to his time in the Premier League at Spurs.
