The 30-year-old also provided two assists as Bayern Munich ran out 7-0 winners in Saturday’s home game at the Allianz Arena.
The Bavarians took an early lead through Eric Maxim Choupo-Moting before Kane doubled the advantage from less than 10 yards out.
Bayern were 4-0 ahead at the break with Matthijs de Ligt and Leroy Sane also hitting the target.
The former Tottenham Hotspur striker completed his brace from the spot in the second-half before adding a late third from close range in the 88th minute.
He also set up Mathys Tel as the home side registered an emphatic 7-0 success to maintain their top position in the standings.
Kane has made a spectacular start to life in the Bundesliga after arriving at the Allianz Arena from Tottenham Hotspur in August.
He scored on his league debut against Werder Bremen and followed that up with a brace in their 3-1 success over Augsburg.
Kane also got on the scoresheet in the 2-2 draw with Bayer Leverkusen after the international break.
He has played in seven matches in all competitions for Bayern, bagging eight goals and four assists.
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