Konrad Laimer opened the scoring inside 10 minutes when he brilliantly flicked a Michael Olise cross beyond Stuttgart goalkeeper Alexander Nubel.
The hosts responded well and had the ball in the back of the net just before the break through Nikolas Nartey but his header was ruled out for offside.
With the lead slender, Bayern manager Vincent Kompany opted for a triple change on the hour mark which included top scorer Kane.
And it took just five minutes for it to pay off with the England captain firing a brilliant strike into the bottom corner from 25 yards.
Fellow substitute Josip Stanisic made it three before Kane scored his second and Bayern’s fourth from the spot after defender Lorenz Assignon was shown a straight red card for a deliberate handball.
The 32-year-old completed his hat-trick in the 87th minute when he got in front of his marker to poke home Olise’s cross from close range.
Kane moves on to 17 Bundesliga goals for the season, nine in front of his closest challenger for the Golden Boot, Eintracht Frankfurt’s Jonathan Burkardt.
Attention turns to the Champions League next for Bayern as they host Portuguese champions Sporting on Tuesday, 9 December (17:45 GMT), while Stuttgart welcome Maccabi Tel Aviv in the Europa League on Thursday (17:45).