Playing in Der Klassiker for the first time, the England captain slotted home from close range in the ninth minute after Dayot Upamecano had headed Bayern in front five minutes earlier, both goals created by Leroy Sane.
Dortmund improved after the break but Bayern's record signing Kane grabbed his second goal, again from close range, to put the game out of reach.
Kane then added the icing on the cake by completing his third hat-trick of the season, and second in succession in the league, in stoppage time after racing clear and finishing coolly.
He is the first player ever to score 15 goals in his first 10 Bundesliga appearances and now tops this season's scoring charts, while also having five Bundesliga assists to his name.
"We are so happy that we have him in the team," captain and goalkeeper Manuel Neuer said, while veteran Thomas Mueller joked that Kane would run out of space for match balls.
"He's soon going to need another hotel room," he said.
After stumbling on the final day of last season to hand Bayern an 11th straight Bundesliga title, Dortmund were itching to make a statement against their rivals and the fabled Yellow Wall of supporters was in full voice before kickoff.
— Harry Kane (@HKane) November 4, 2023
Bayern were humiliated 2-1 by third-tier Saarbruecken in the German Cup second round in midweek and, despite Kane's goals, they have not been firing on all cylinders, though they remain unbeaten in the league.
However, they produced their best display of the season to outclass Dortmund, seizing control from the first whistle.
Upamecano escaped his marker to thump a close-range header into the net from a delivery by Sane, who was unstoppable early on and linked brilliantly down the left before sliding a low cross for Kane to tap home the visitors' second goal.
Kane also hit the post as Bayern dominated although Dortmund's Donyell Malen wasted a great chance to pull a goal back in first-half stoppage time, curling his shot over the bar.
The home side regained some control in the second half without really offering much threat.
Kane scored his second goal from Kingsley Coman's pass and was then played through by Aleksandar Pavlovic's pass to finish past keeper Gregor Kobel, sealing Bayern's biggest winning margin away at Dortmund since a 5-1 victory in 2009.
The former Tottenham striker now has 17 goals in his first 14 Bayern games in all competitions.
Additional reporting by Reuters.