Kane, who was on target in his league bow at Werder Bremen last weekend, converted a first half penalty on Sunday and added another goal after the break.
The win sees Bayern with a maximum six points from two games and second in the table behind Union Berlin who also have two wins.
"A really good win," Kane, 30, told DAZN after the match.
"The first league game in our stadium -- I wanted to make sure we got the three points."
Augsburg defended well under relentless Bayern pressure but they went behind on 32 minutes when Felix Udoukhai put through his own goal.
A VAR intervention found an Augsburg handball just inside the box minutes later, bringing Kane to the spot.
Kane, who has shown no signs of being burdened by his 100-million euro ($110 million) transfer fee, blasted the ball nervelessly down the middle to double Bayern's lead.
Bayern manager Thomas Tuchel toasted a "good win, a clear win, a deserved win" but said his side still had "real potential to improve."
Augsburg manager Enrico Maassen admitted victory was out of reach for his "unlucky side" when "we gave away two goals and went into halftime 2-0 down."
Bayern came into the clash having lost their last two competitive matches at their Allianz Arena home -- both against RB Leipzig in the Bundesliga last season and this campaign's Super Cup.
They struggled to make an impression early on against Augsburg who had won two of their past four games against the Bavarian giants.
The German champions needed a slice of luck to take the lead and got it midway through the opening half.
That paved the way for Kane to grab the headlines, tucking away a penalty and then showing his versatility in front of goal in the second half, chipping a first-touch finish into the goal for a 3-0 lead.
Augsburg grabbed a late consolation goal through Dion Beljo with four minutes remaining.
- Marmoush strikes late -
Eintracht Frankfurt forward Omar Marmoush equalised in injury time to snatch a point in a 1-1 draw at derby rivals Mainz.
With the clock winding down, the home side were in front after Lee Jae-sung's first-half header, but Frankfurt's Junior Dina Ebimbe dribbled past three Mainz defenders and cut back to give Marmoush a simple tap in to level the scores.
Frankfurt had been reduced to 10 men after winger Ansgar Knauff had kicked Leandro Barreiro in the face as he challenged for the ball with 30 minutes remaining.
The dismissal galvanised the visitors and Marmoush's goal, the Egyptian's first since moving from Wolfsburg in the summer, ensured they remain unbeaten this season.
Linked with a big money move to Paris Saint-Germain, Frankfurt striker Randal Kolo Muani looked the man most likely to open the scoring, going close twice in the first 15 minutes.
Defeated 4-1 at Union Berlin in their opening game last weekend, Mainz shook off a tentative start and looked to have won a penalty for handball with 23 minutes gone, but the decision was overturned by VAR.
The visitors were on the board just two minutes later however, Lee Jae-sung pouncing on a mistake from Frankfurt goalkeeper Kevin Trapp to head his side into the lead.
Trapp rose high to collect the ball but lost possession after colliding with teammate Robin Koch, with Lee heading the ball in from close range.
Knauff's dismissal looked to have Mainz on track for victory before Marmoush's late intervention saw the spoils shared.