The Poland striker had gone five league matches without finding the net but scored either side of half-time.
Lewandowski put Bayern ahead when he latched on to Arjen Robben's clever pass before he turned provider with a low cross for the Dutch winger to net.
Robben's intelligent through ball allowed Lewandowski to score his second before Ja-Cheol Koo's consolation goal.
Bayern's victory was emphatic although Carlo Ancelotti's side benefitted from some poor defending by the home side.
It was Bayern's second Bavarian derby success over Augsburg in the space three days following a 3-1 win in the German Cup on Wednesday.
Bayern, the defending Bundesliga champions, remain two points clear of second-placed RB Leipzig, who claimed a 2-0 away win over Darmstadt.
Austria midfielder Marcel Sabitzer secured the points for Leipzig with two goals in the second half.
Match ends, FC Augsburg 1, FC Bayern München 3.
Second Half ends, FC Augsburg 1, FC Bayern München 3.
Attempt saved. Robert Lewandowski (FC Bayern München) right footed shot from the right side of the box is saved in the bottom right corner. Assisted by Joshua Kimmich.
Offside, FC Augsburg. Philipp Max tries a through ball, but Dong-Won Ji is caught offside.
Corner,  FC Augsburg. Conceded by Holger Badstuber.
Attempt blocked. Martin Hinteregger (FC Augsburg) left footed shot from the right side of the six yard box is blocked. Assisted by Paul Verhaegh with a cross.
Corner,  FC Augsburg. Conceded by David Alaba.
Attempt blocked. Robert Lewandowski (FC Bayern München) header from the centre of the box is blocked. Assisted by Philipp Lahm with a cross.
Foul by Thiago Alcántara (FC Bayern München).
Konstantinos Stafylidis (FC Augsburg) wins a free kick in the defensive half.
Douglas Costa (FC Bayern München) wins a free kick in the attacking half.
Foul by Paul Verhaegh (FC Augsburg).
Douglas Costa (FC Bayern München) wins a free kick in the attacking half.
Foul by Jan Morávek (FC Augsburg).
Joshua Kimmich (FC Bayern München) wins a free kick in the defensive half.
Foul by Philipp Max (FC Augsburg).
Substitution, FC Augsburg. Jan Morávek replaces Gojko Kacar.
Attempt missed. Philipp Lahm (FC Bayern München) left footed shot from the centre of the box is close, but misses to the right. Assisted by Douglas Costa.
Corner,  FC Bayern München. Conceded by Georg Teigl.
Xabi Alonso (FC Bayern München) wins a free kick in the defensive half.
Foul by Ja-Cheol Koo (FC Augsburg).
Substitution, FC Bayern München. Joshua Kimmich replaces Arjen Robben.
Georg Teigl (FC Augsburg) is shown the yellow card.
David Alaba (FC Bayern München) wins a free kick in the defensive half.
Foul by Georg Teigl (FC Augsburg).
Offside, FC Bayern München. Manuel Neuer tries a through ball, but Douglas Costa is caught offside.
Attempt missed. Gojko Kacar (FC Augsburg) right footed shot from outside the box misses to the left. Assisted by Ja-Cheol Koo with a headed pass.
Attempt missed. Mats Hummels (FC Bayern München) header from the centre of the box is close, but misses to the right. Assisted by Douglas Costa with a cross following a corner.
Attempt saved. Mats Hummels (FC Bayern München) header from the centre of the box is saved in the centre of the goal. Assisted by Arjen Robben with a cross.
Corner,  FC Bayern München. Conceded by Konstantinos Stafylidis.
Robert Lewandowski (FC Bayern München) wins a free kick in the defensive half.
Foul by Christoph Janker (FC Augsburg).
Corner,  FC Bayern München. Conceded by Ja-Cheol Koo.
Attempt saved. Mats Hummels (FC Bayern München) header from the centre of the box is saved in the bottom left corner. Assisted by Arjen Robben with a cross.
Corner,  FC Bayern München. Conceded by Konstantinos Stafylidis.
Goal!  FC Augsburg 1, FC Bayern München 3. Ja-Cheol Koo (FC Augsburg) right footed shot from the centre of the box to the bottom right corner. Assisted by Daniel Baier.
Hand ball by Douglas Costa (FC Bayern München).
Foul by Thiago Alcántara (FC Bayern München).
Daniel Baier (FC Augsburg) wins a free kick in the defensive half.
Substitution, FC Augsburg. Georg Teigl replaces Halil Altintop.
Robert Lewandowski ended his Bundesliga goal drought to help leaders Bayern Munich ease to victory over Augsburg.