Bayern’s celebrations looked in danger of being delayed when the ball went into their net off their own defender Niklas Suele but Corentin Tolisso and James Rodriguez replied for the Bavarians before halftime.
Arjen Robben and Sandro Wagner then put the finishing touches in the second half.
Bayern, who have 72 points from 29 games, moved 20 ahead of Schalke 04, who have six games to play including Saturday evening’s visit to lowly Hamburg SV.
Augsburg dominated the first half-hour and took an 18th-minute lead with one of the oddest goals Bayern have conceded for a long time.
Defender Jerome Boateng, under no pressure, lost control of the ball and was dispossessed by Sergio Cordova leaving the Venezuelan with only Sven Ulreich to beat.
Cordova’s shot was blocked by the Bayern goalkeeper but ricocheted off the hapless Suele’s face and into the net.
Bayern equalised out of the blue when Tolisso was left unmarked to head in from Joshua Kimmich’s cross in the 32nd minute and the Bavarians took the game by the scruff of the neck.
Rodriguez slipped the ball home from Juan Bernat’s clever backheel six minutes later and the Colombian also set up Robben for the third when he hooked the ball back from the touchline just after the hour.
Bayern, who have won 28 German titles altogether, had eased up by then but there was still time for Sandro Wagner to score a fourth in the 86th minute.
Elsewhere, Bruno Labbadia claimed his first win since taking over at VfL Wolfsburg when they beat Freiburg 2-0 away with two goals from Daniel Didavi.
The win moved the Wolves two points clear of Mainz 05 who stayed in the relegation playoff spot after drawing 1-1 at Cologne.
Jonas Hector gave Cologne an early lead but Pablo de Blasis levelled five minutes into the second half, leaving the Billy Goats six points adrift of safety in 17th place.
Thorgan Hazard scored two second-half goals, the second from a penalty, to give Borussia Moenchengladbach a 2-1 win over Hertha Berlin after Salomon Kalou had put the visitors in front.
It was a remarkable personal achievement for Bayern’s 72-year-old coach Jupp Heynckes.
He came out of retirement in October to replace Carlo Ancelotti for his fourth stint at the club with Bayern five points off the pace.