Last season's treble winners 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 creditably 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 rivals 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 were without Dominik Reinhardt and Panagiotis Vlachodimos due to tendon problems.
Bayern immediately pushed their 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 swivelled to thrash a left-footed shot into the top corner.
Pep Guardiola's side almost doubled their advantage in the 16th minute when Ribery's shot from the left-hand edge of the box skidded past the far post as Mario Gotze just failed to find the decisive touch.
Augsburg had started to feel their way into the game, the industrious Andre Hahn at the centre of their good play, but when Paul Verhaegh brought down Ribery 20 yards out three minutes before the break, the France international punished them with an unstoppable free-kick.
Bayern looked to press home their 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 visitors' 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 penalised for handling Robben's cross.
Bayern now turn their attentions towards the eagerly anticipated trip to Dortmund after the international break, and do so from a position of strength.