The Germany international was on target twice as Bayern maintained its fine run of form and kept pace with leader Dortmund ahead of Wednesday's table-top clash
It took just 24 seconds for Gomez to open the scoring, but Koo Ja-Cheol drew the visitors level on 15 minutes. Entering the final half-hour, the Bayern striker completed his double, putting the hosts ahead for good.
Moments after kick-off, Franck Ribery raced up the left flank and crossed to the wide-open Gomez, whose initial volley was saved by Simon Jentzsch before his follow-up shot finally beat the goalkeeper.
Augsburg reacted well defensively, and was unlucky not to equalize on the quarter-hour mark as a Bayern defensive gaffe saw the ball run through to Marcel Ndjeng, who had a close-ranged effort cleared off the line by Jerome Boateng.
And on 23 minutes, FCA was rewarded for their positive play as they restored parity. Axel Bellinghausen slipped behind the hosts' defense before cutting back a low cross to Koo, who coolly nutmegged Manuel Neuer and into the net.
Ribery came close to restoring the hosts' lead shortly after with a well-struck shot from distance, and Gomez and Bastian Schweinsteiger also had good chances before halftime, but missed the mark as the two sides entered halftime on level ground.
Bayern came out stronger in the second half, and within minutes, Toni Kroos was brought on for Anatoliy Tymoshchuk to partner Schweinsteiger in defensive midfield. Jupp Heynckes' tactical shift bore fruit as his side heaped pressure on the visitors.
Gomez finally broke the deadlock on the hour mark following a sparkling bit of play from Philipp Lahm, who played Arjen Robben past the FCA defense. The Dutchman squared for Gomez, who tapped home from close range.
Gomez's second goal seemed to take the steam out of Augsburg's sails, and the visitors were not properly staffed to chase the game, with striker Nando Rafael having been replaced by defensive midfielder Jan-Ingwer Callsen-Bracker early in the second half.
Bayern's chances were limited, but the hosts had little trouble preserving their lead. And entering the final 10 minutes, the hosts were closer to adding a third than conceding the equalizer. Schweinsteiger had a header saved, and Ribery flashed a dangerous effort just wide of the right post.
In injury time, FCA made one final push to extend its unbeaten streak to seven games, but ultimately fell short. It still stands in 14th place, currently clear of the relegation zone.
The result leaves Bayern three points behind leaders Dortmund entering the final five matches of the season. The German record champions will have the chance to claim the lead on Wednesday, as they square off with the leaders at the Westfalenstadion.