Bayern were frustrated for long spells of the match at the Allianz Arena, but Kane's late double - both goals coming in the final 10 minutes - ultimately sealed a 2-1 triumph that made it four wins from four and secured their place in the knockout rounds with two games to spare.
Galatasaray did give themselves some late hope when Cedric Bakambu pulled one back in the third minute of stoppage time, but there was no time for a dramatic late equaliser.
Kane was coming into the match having scored hat-tricks in his last two appearances, and he almost made an early start towards another when he stung the palms of Fernando Muslera from range after five minutes.
Muslera was called into action again to thwart Leroy Sane when one on one on the half-hour mark, before Manuel Neuer did the same to deny Mauro Icardi at the other end shortly before half time.
Kane came close again eight minutes into the second half when he could only divert the ball against the post at full stretch after Kingsley Coman's dangerous cross had been glanced on by Leon Goretzka.
Galatasaray had the ball in the back of the net not long after, only for Baris Yilmaz's goal to be disallowed for offside, but the visitors looked well on course for an important point as the clock ticked in to the final 10 minutes.
Kane has already underlined his status as a difference-maker during his short time in Bavaria, though, and he came up trumps again for the home side with a quickfire brace in the space of six minutes.
The England captain's first - a glancing header from Joshua Kimmich's free kick - was initially ruled out for offside, but VAR overruled the onfield decision after replays showed that Kane was clearly onside when the ball was played.
Kane then doubled his personal tally on the night by putting a simple finishing touch to a slick passing move involving Thomas Muller and Mathys Tel, making it 19 goals in just 15 appearances for Bayern since joining in the summer.
Muller missed a glorious chance for a third when he sent a dinked finish spinning agonisingly wide from inside the area, and that could have proved costly as Galatasaray soon pulled one back, Bakambu latching on to a simple pass over the top before steering his effort across Neuer and into the bottom far corner.
Bayern safely held on though, making it into the last 16 for a 20th time and joining Manchester City, Real Madrid and Inter Milan in the knockout rounds, with the latter two also stamping their ticket on Wednesday night.
Galatasaray, meanwhile, slip down to third in Group A following Copenhagen's dramatic 4-3 win over Manchester United, although just one point now separates the other three teams in the table.