The Bavarian powerhouse dominated with the exception of a 15-minute spell after halftime when Antonio Nusa went close for the visitors in Munich on Friday (local time).
Nusa, who also had a goal ruled out, served to only rile more goals from Kane, who was supported by the outstanding Michael Olise, who scored two, and Luis Diaz, who got the other goal and set up two on his Bundesliga debut.
"We wanted to come out with a statement and that's exactly what we did," Kane told Sky TV. "Leipzig are a good side, but today we were on top in all areas and we were clinical when we needed to be, so we can enjoy this."
Olise broke the deadlock in the 27th minute and Diaz had the home fans celebrating again five minutes later when he rifled the ball in off the underside of the crossbar after a well-worked move involving Kane and Serge Gnabry, who laid the ball off for Diaz with his heel.
Olise got his second goal two minutes after that, Gnabry easing past two Leipzig defenders to set up the French winger.
Leipzig improved after the break, no doubt after a stern talk at halftime from new coach Ole Werner, one of six coaches making their Bundesliga debuts with new clubs this weekend.
But Kane got Bayern's fourth in the 64th minute, sending a defender flying after taking Diaz's pass, then tucking the ball past Peter Gulacsi in the Leipzig goal.
Bayern coach Vincent Kompany made four changes at once in the 68th, but there was no respite for Leipzig.
Kane got his second goal in the 74th minute - again set up by Diaz - and the England captain completed his first hat-trick of the season four minutes later.
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