Jerome Boateng, Franck Ribery and Thomas Muller grabbed the goals for the defending champions, who never allowed an outclassed Augsburg into the game.
The win saw Bayern - who last lost a league game in October 2012 - beat Hamburg's 36-match unbeaten run which was set between January 1982 and January 1983.
Dortmund went ahead in first-half injury time through Marco Reus but Ricardo Rodriguez equalised in the 56th minute and Ivica Olic gave the hosts the win with the second after 69 minutes.
On Sunday, Mainz piled the pressure on local rivals Eintracht Frankfurt with a last-gasp 1-0 win at the Coface Arena.
Eric-Maxim Choupo-Moting scored in the 88th minute of the Rhine-Main derby, earning Thomas Tuchel's men only their second win in their past nine Bundesliga games and extending Frankfurt's winless run to seven.
Teenager Timo Werner scored twice as Stuttgart secured an important 3-1 victory at Freiburg to stay close to the European places.
Vedad Ibisevic gave Stuttgart the lead in the ninth minute and the 17-year-old Werner made it 2-0 just 83 seconds later.
Freiburg registered their goal in the 78th minute through Mike Hanke, but Werner struck again in the 82nd minute.
Back to Saturday's action and Heung-Min Son came back to haunt his former club as his first ever Bundesliga hat-trick helping Bayer Leverkusen to a 5-3 victory over Hamburg.
The South Korean, who switched clubs in the summer, helped the hosts race into a 2-0 lead, but Maximilian Beister and Pierre-Michel Lasogga brought Hamburg back on terms.
Son's third restored Leverkusen's lead before Stefan Kiessling gave them a cushion once again.
Lasogga brought Hamburg back into the game before Gonzalo Castro sealed victory for Sami Hyypia's men, taking them back level on points with second-placed Dortmund.
Kevin-Prince Boateng scored two goals as Schalke came from a goal down to beat Werder Bremen 3-1.
Bremen went ahead in the 22nd minute through Felix Kroos before Boateng struck in the 64th and 85th minutes with Jefferson Farfan sealing victory a minute into injury time.
Adrian Ramos scored twice and set up a third as Hertha Berlin gave up a two-goal lead but still won 3-2 at Hoffenheim.
Anis Ben-Hatira gave Hertha the lead in the 13th minute and Ramos doubled the advantage after 53 minutes.
Sejad Salihovic struck a penalty in the 70th minute and added a free-kick in the 81st as Hoffenheim pulled level, but Ramos popped up with Hertha's winner in the 84th minute.
In the late game, Nurnberg were left cursing their luck in a 3-1 defeat to Borussia Monchengladbach.
Josip Drmic gave Nurnberg the lead in the 21st minute but Juan Arango's strike and a Niklas Stark own goal gave the hosts the lead.
However, controversy struck in the 80th minute when Drmic thought he had brought Nurnberg level, only to have what seemed to be a legitimate goal chalked off after the officials failed to spot that the ball had crossed the line.
Patrick Herrmann compounded Nurnberg's misery with a third goal late on.
On Friday night, Hannover and Eintracht Braunschweig delivered only the second goalless draw of the Bundesliga season in a Lower Saxony derby which proved to be an anti-climax.
The anticipation was high of a game which was last played in 1976, but in spite of Hannover's attacking efforts, Braunschweig managed to get away with precisely what they had come for - a point.