Despite having 80 per cent possession and 13 shots to the visitors' two in the first half, Bayern were somewhat sluggish on their return from the international break at Allianz Arena.
Jamal Musiala was the home team's brightest spark and was repeatedly denied by an in-form Nediljko Labrovic in Augsburg's goal.
It looked like it would be a night of frustration for Vincent Kompany's side but substitute Mads Pedersen inexplicably handled in the box and Kane emphatically dispatched the resulting spot-kick in the 63rd minute.
The Fuggerstadter posed little threat to Manuel Neuer's goal, whereas the Bavarians peppered Labrovic's - but the Croatian was up to the task.
The England captain, who was fairly anonymous for the majority of the contest, bagged his second penalty in the 93rd minute after Keven Schlotterbeck fouled the 31-year-old in the area - with the defender then sent off for a second yellow.
The former Tottenham star then made it 3-0 two minutes later when he beautifully controlled Leon Goretzka's cross and slotted home to put Bayern eight points clear of second-placed RB Leipzig on Friday night.