Two goals in the space of a minute just before the hour mark set the visitors on their way during a superb second-half performance, which helped stretch their lead at the top of the table to 10 points.
Mehdi Benatia opened the scoring with a flicked header, before Robben curled home just a few seconds later.
A powerful drive from Robert Lewandowski extended the lead after 68 minutes, before the imperious Robben rounded off another eye-catching performance by slotting home the fourth after some good work from Sebastian Rode.
The win sees the champions claim a 16th victory in their last 19 Bundesliga outings, while also bringing an emphatic end to Augsburg's four-match winning streak.
A spirited performance from the hosts was enough to stifle Bayern's attacking talents during a tightly contested first half, but Augsburg rarely looked like troubling their Bavarian neighbours.
Despite having no fresh injury concerns, Bayern boss Pep Guardiola made five changes to the side that beat CSKA Moscow in the UEFA Champions League in midweek, but his team showed no sign of surrendering any momentum during the opening stages.
A confident start saw the champions dominate possession in the first 10 minutes, while star wingers Franck Ribery and Robben both looked lively, with the latter firing wide twice in the space of two minutes.
Although the visitors looked the more likely to break the deadlock, they were given a scare after 15 minutes when a timely interception allowed Hong Jeong-ho to find Nikola Djurdjic in the area, but a poor first touch saw the chance go begging.
Augsburg, who made two changes from their win over Cologne last week, came into this game in good form and they appeared to grow in confidence as the first half progressed, although they still failed to really test Manuel Neuer.
Bayern struggled to rediscover their earlier rhythm as the game approached the half hour but still looked the more threatening, Lewandowski seeing a close-range effort well blocked by Ragnar Klavan after 27 minutes, before Robben forced a superb save from goalkeeper Alex Manninger with a powerful angled drive inside the area.
Ribery also continued to cause problems and nearly had a hand in opening goal of the game a minute after the break, bursting down the left before whipping a cross towards Lewandowski, who headed over the crossbar from 10 yards.
Xabi Alonso forced an acrobatic save from Manninger with a header of his own four minutes later, before Ribery somehow contrived to fire wide after being played in by Bastian Schweinsteiger's flick on.
However, the Frenchman made amends just before the hour, curling in a superb corner that was met by the head of Benatia, who flicked the ball onto the post and into the net.
The lead was doubled a minute later, when Robben carved out some space for himself on the edge of the box before curling a wonderful dipping strike in off the woodwork.
The floodgates then opened, as Lewandowski found a third 10 minutes later with a low drive from the edge of the area, before Robben capped off a brilliant performance by sealing his second of the afternoon.
Some excellent work down the left allowed Rode to find Robben in the area, allowing the Dutchman to slot home and cap off a magnificent performance.