The England striker netted all three goals in the second half, the first two from the penalty spot and the third with a fine touch and header deep into stoppage time, to bring up his half-century of goals in Germany's top division in just 43 games.
"It was somehow typical Harry Kane," Bayern coach Vincent Kompany said. "It was close in the first half, close, close, close, and then it happens, I think three goals in 15 minutes. Of course, he can do that.
"But for such a player, I have to say, he had a lot of chances that he could maybe make more of. Then, in an instant, everything is perfect and he can score many goals."
Kane scored 36 goals in 32 league appearances last season, his first for Bayern after joining from Tottenham. It was also the first season since 2012 that Bayern failed to win the title, as Bayer Leverkusen went undefeated to snap Bayern's winning run.
As well as 14 league goals this season, Kane has scored five in the Champions League and one in the German Cup.
Friday's win saw Bayern open up an eight-point gap at the top with their sixth straight clean sheet across all competitions.
The Bavarians, who take on Paris Saint-Germain in the Champions League on Tuesday, are undefeated in the Bundesliga and on 29 points following their fifth consecutive league game without conceding a goal.
Second-placed RB Leipzig, who travel to Hoffenheim on Saturday, are on 21 points.
"We said at half time we should just keep going and create chances," Kane told reporters. "Thankfully we got the penalty and from there the game opened up."
Bayern had possession and chances, but the visitors defended resolutely with Augsburg goalkeeper Nediljko Labrović denying Kane, then twice Jamal Musiala to keep the game scoreless at half-time.
Bayern counterpart Manuel Neuer, who overcame a rib injury to start, had little to do at the other end.
Then Mads Pedersen was penalised for handball following a VAR review and Kane duly broke the deadlock in the 63rd.
Bayern continued as before with 80% possession, but had to wait for Keven Schlotterbeck to be penalised through VAR for a foul on Kane.
Kane sealed the result in the third minute of stoppage time and there was still time for him to grab another.
It's Bayern's seventh consecutive win since Oct. 23 when they lost at Barcelona 4-1 in the Champions League.
"We have a big week ahead with PSG in the Champions League then Dortmund then Leverkusen in the Cup. We're in a good moment, we're feeling good and should just keep the momentum up," Kane said.
Information from The Associated Press and Reuters contributed to this report.