Gomez scored with a header from close range in the 15th minute, when he was well placed to meet Mohamed Amsif’s save from Daniel van Buyten’s header.
Ribery twisted inside Dominik Reinhardt before slotting past the helpless Amsif to make it 2-0 in the 28th.
Hajime Hosogai pulled one back for Augsburg in the 59th when Bayern failed to clear a corner, the ball falling kindly for the Japan midfielder to score from close range. David Alaba’s desperate lunge failed to stop the ball crossing the line.
Manuel Neuer denied Augsburg an equalizer in the 83rd, with a great reflex save from Edmond Kapllani who was through one-on-one.
In a nervous finish for the Bundesliga leader, Anatoliy Tymoshchuk received a straight red card for a dangerous tackle on Daniel Baier in injury time.
Earlier, Finland striker Teemu Pukki scored twice in his first Bundesliga start to salvage a 2-2 draw for Schalke at Hannover on Sunday.
Pukki opened the scoring in the 26th minute and tied the game in the 73rd, after an own goal from Kyriakos Papadopoulos in the 29th and Mohammed Abdellaoue’s header in the 59th had given Hannover the lead.
“Schalke were the better team in the end, with two good chances,” said Hannover coach Mirko Slomka. “It’s not good that we’re conceding so many chances. That didn’t please me at all.”
Schalke missed the chance to go second and dropped to fifth, three points ahead of Hannover.
Klaas Jan Huntelaar failed to recover from a broken nose sustained in the Europa League on Thursday, giving Pukki his opportunity in the Dutchman’s place.
“I really wanted to show what I can do. I’m really happy,” Pukki said.
The 21-year-old outpaced Karim Haggui for Christian Fuchs’ ball forward for his first Bundesliga goal, taking it past the outrushing Ron-Robert Zieler with his first touch and slotting into the empty net.
Hannover equalized three minutes later when Papadopoulos’ attempted clearance of Lars Pander’s free kick ended in his own net.
Abdellaoue was denied a penalty before the interval, when he went down under Papadopoulos’ challenge.
Pukki went agonizingly close when his shot whizzed by the post in the 57th, moments before Didier Ya Konan went just as close at the other end.
Abdellaoue scored two minutes later, when Jan Schlaudraff crossed brilliantly with the outside of his foot for the Norway striker to head in his eighth goal of the season.
Pukki scored his second from a narrow angle after a great run from Jefferson Farfan, beating Zieler with a powerful shot across goal.
“In the end we had so many chances to score the winner, but it’s still a good result against Hannover,” Pukki said. “We believe we can (reach the Champions League). We can go far this season.”