The Frenchman struck after Jerome Boateng's early goal and a last-gasp penalty from Thomas Muller as Bayern made it 37 games without defeat in the league.
Last season's treble winner surpassed the mark set by Hamburg in 1983 to reach 37 games unbeaten thanks to excellent first-half finishes from Jerome Boateng and Franck Ribery and a stoppage time penalty from substitute Thomas Muller.
Augsburg performed well in the early stages of the second period but failed to extend Germany number one Manuel Neuer in the Bayern goal.
The afternoon was made sweeter for home fans at the Allianz Arena when news filtered through of a 2-1 defeat for title rival Borussia Dortmund at Wolfsburg, meaning Bayern's advantage at the Bundesliga summit is extended to four points.
Arjen Robben returned from a groin complaint to be named among the Bayern substitutes, while Augsburg was without Dominik Reinhardt and Panagiotis Vlachodimos due to tendon problems.
Bayern immediately pushed the visitors back and, in the third minute, Augsburg goalkeeper Marvin Hitz dashed out of his area to thwart Mario Mandzukic before scampering back to touch Javi Martinez's speculative follow up behind.
Replays suggesting Mandzukic was offside when challenging Hitz for the loose ball leant an air of controversy as Bayern scored from the resulting corner – Dante heading on towards the back-post where Boateng swiveled to thrash a left-footed shot into the top corner.
Pep Guardiola's side almost doubled the advantage in the 16th minute when Ribery's shot from the left edge of the box skidded past the far post as Mario Gotze just failed to find the decisive touch.
Augsburg had started to feel its way into the game, the industrious Andre Hahn at the center of the good play, but when Paul Verhaegh brought down Ribery 20 yards out three minutes before the break, the France international punished the club with an unstoppable free kick.
Bayern looked to press home the advantage early in the second period but Ribery and Gotze were guilty of over-elaborating on the final pass.
In the 52nd minute, Mandzukic managed to find Ribery in the area but he fired his attempt wide.
Augsburg boss Markus Weinzierl then replaced Halil Altintop with Arkadiusz Milik and the new man might have made an instant impact when David Alaba blocked his shot from Hahn's lofted pass.
That move heralded the visiting side's most productive spell of the match, Hahn dragging wide from 18 yards in the 64th minute, but Bayern could have made the points safe when Martinez was left unmarked to head wide from a Gotze corner before the latter made way for Robben.
The Dutch winger produced a lively cameo and twice game close to finding the net, but it was fellow replacement Muller who confidently converted from 12 yards after Matthias Ostrzolek was harshly penalized for handling Robben's cross.
Bayern now turns its attention towards the eagerly anticipated trip to Dortmund after the international break, and does so from a position of strength.