The win clinched the domestic double for Bayern for the first time in six years.
Kane headed home for a 55th-minute lead, then bagged his second after a superb move 10 minutes from time before earning his third goal on Saturday evening with a stoppage-time penalty.
The England captain, in stellar form heading into next month's World Cup, is only the third player ever to score in every German Cup round, including the final.
He has also netted 61 times in 51 matches across all competitions for Bayern this season.
"Maybe a bit of patience," Kane said when asked how the game turned. "The first half was tough. They like the man-to-man pressure. But they started to tire a little bit.
"In the second half we were a bit more patient with the ball, a bit more quality. We deserved to win.
"A record-breaking season. We wanted to win today. We got the job done."
Bayern honorary president and club supremo Uli Hoeness called Kane's transfer the best the club has ever made.
"To have those types of words, knowing how many great players have played here, it's special," Kane said.
The Bavarians, who were eliminated in the Champions League semi-finals by Paris St Germain this month, missing out on a potential treble, were on the back foot in the first half.
They struck against the run of play when Kane, who topped the Bundesliga scorers' list for a third straight season, headed in a perfect Michael Olise assist 10 minutes after the restart.
The match was briefly interrupted on the hour after both sets of fans lit flares in the stands, with smoke trapped under the sprawling Olympic Stadium's roof limiting visibility.
Stuttgart's Chris Fuehrich came close with a 76th minute header but it was Kane who delivered a masterclass finish four minutes later to end their hopes of a comeback.
The 32-year-old first rattled the crossbar with a powerful shot and then received a pass from Luis Diaz on the rebound in the box, turned quickly to shake off one marker and beat goalkeeper Alexander Nuebel with a low drive.
Kane capped his superb performance with a stoppage-time spot kick to round off a perfect night for the striker.
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