Amid a smattering of snow and plunging temperatures, the hosts were not at their flying best against their Bavarian rivals.
But Kane's perfect 63rd-minute penalty, after a clear handball from substitute Mads Pedersen, had the hosts ahead and another spot-kick in stoppage time followed by an easy header after his own flick gave him 14 goals in 11 league games.
Vincent Kompany's side top the table by eight points ahead of second-placed RB Leipzig going to lowly Hoffenheim and new coach Christian Ilzer in Saturday's main program.
Bayern, who lost out on the title to Bayer Leverkusen last term for the first time since 2012, were boosted by goalkeeper Manuel Neuer declaring himself fit after a rib problem and he had little to do.
Leon Goretzka made another rare start and fired just wide with the first decent chance.
On a new pitch at the Allianz Arena following NFL exertions, Bayern stayed calm and Joshua Kimmich, Jamal Musiala and Kingsley Coman all went close before the break.
The hosts needed to up the pace after the interval and Pedersen's reckless handball, picked up on video review, opened the door.
A second penalty thanks to VAR saw Keven Schlotterbeck sent off for a second booking after tripping Kane, before the Bayern striker converted.
A fourth hat-trick already in all competitions this term for England captain Kane soon followed. He has now scored 25 penalties in a row - with his miss in the 2022 World Cup quarter-final defeat to France a rare blemish.
"We knew it was going to be difficult to break them down at times," Kane told DAZN. "Thankfully we got the penalty to open the game up and from there we did well to kill the game off."
Augsburg, still without an away win this term, remain 13th.
Bayern travel to old foes Borussia Dortmund next weekend after hosting Paris Saint-Germain in the Champions League on Tuesday.
Other Saturday games on matchday 11 include fourth-placed champions Leverkusen at home to Heidenheim.
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