After a Champions League hat-trick against Napoli in midweek, Gomez headed home on 16 minutes for his 13th Bundesliga goal of the season - his 20th in 18 games in all competitions - to go top of the league's scoring charts.
But Bayern did not have it easy in the Bavarian derby and victory came at a price as midfielder Anatoliy Tymoshchuk was sent off in the 93rd minute for a reckless tackle on Augsburg's Daniel Baier.
"In the first half, we played well, but didn't play with such authority after the break," admitted Munich midfielder Toni Kroos. "Our aim must be to play with confidence and dominate our opponents."
Having lost vice-captain and Germany star Bastian Schweinsteiger with a broken collarbone on Wednesday, Bayern coach Jupp Heynckes opted to start with Austria teenager David Alaba for 'Schweini' in midfield.
After Gomez put the guests ahead, France winger Franck Ribery doubled Bayern's lead when he fired home on 28 minutes.
But Augsburg refused to roll over and Bayern conceded only their fourth league goal this season when Japan's Hajime Hosogai managed to fire his shot over the line. Although Alba cleared, the referee awarded the goal.
"We gave everything right until the end. We deserve respect for that," said Augsburg coach Jos Luhukay.
Hosogai's 59th minute goal made for a tense final half-hour and Germany goalkeeper Manuel Neuer made a crucial save with seven minutes to go after Augsburg substitute striker Edmond Kapllani broke through the defence.
"He did his job and paid back a million," quipped Bayern president Uli Hoeneß after Neuer cost Bayern €22 million euros from rivals Schalke in the summer.
Bayern held on to claim the three points and re-establish their lead after the chasing pack of Dortmund, Bremen and Mönchengladbach had all won Saturday.
Earlier, Finland striker Teemu Pukki scored twice on his first full appearance for Schalke 04 to earn a point for his side in their 2-2 draw with Hannover 96 as the Royal Blues remain fifth in the table with Hannover sixth.
On Saturday, defending champions Borussia Dortmund went second with a 5-1 hammering of VfL Wolfsburg as Japan star Shinji Kagawa and Germany teenager Mario Götze ran riot.
Kagawa set up both the first and fourth goals for Dortmund and netted one of his own, while Götze scored twice.
The win came at a price, however, as defender Neven Subotic is expected to miss the next six weeks, including Champions League games against Arsenal and Marseille, with a facial fracture which required surgery.
Werder Bremen went third, and joined Dortmund and Gladbach on 23 points, when they came back from 2-0 down to win 3-2 against ten-man Cologne thanks to a second-half hat-trick from Peru's Claudio Pizarro.
Mönchengladbach moved up to fourth after their come-back win at Hertha Berlin as Bayern target Marco Reus scored both their goals in a 2-1 win.
Bayer Leverkusen threw away a 2-0 lead at home as strugglers Hamburg roared back with goals from ex-Germany players Marcell Jansen and Heiko Westermann to leave their side 16th in the league.
Hosts Nuremberg lost 2-1 at home to Freiburg as Senegal striker Papiss Cisse converted a penalty with the last kick of the game for the visitors which moves them to up third from bottom as mid-table Hoffenheim drew 1-1 with Kaiserslautern.