Gomez scored his league-high 13th goal after 16 minutes and Ribery doubled the lead in the 28th, but Hajime Hosogai pulled a goal back in the 59th, and Neuer saved Edmond Kapllani's breakaway in the 83rd minute to preserve Bayern's win.
Bayern extended its lead atop the standings to five points over three teams on 23 points, including defending champion Borussia Dortmund. Augsburg lost for the third time in its last five, and is 20 points off the pace in last place.
Gomez got Bayern off to a fast start off a corner from Ribery that was turned on goal from the near side by Daniel van Buyten. Although Augsburg goalkeeper Mohamed Amsif knocked the shot down, Gomez headed the rebound in from just off the line.
Even without midfield general Bastian Schweinsteiger, lost in the midweek win over Napoli in the Champions League to a broken collarbone, Bayern looked in complete control when it put a second goal past Amsif inside 30 minutes.
Thomas Muller played Ribery into the area on the left, and the Frenchman made a nice cut to his left to lose two defenders, then sliced a left-footed shot to the far post for his sixth goal of the season.
Augsburg did not go away silently, as Hosogai pulled a goal back just before the hour mark when he hammered a shot over the line that sneaked just into the goal before David Alaba cleared.
The struggling hosts were denied a point when Kapllani breakaway shot aimed at the left post was deflected away by the right fingertips of Neuer, who charged off his line to pressure the shot and made his biggest save of the match.
Anatoliy Tymoshchuk received his marching orders in stoppage time but Bayern's ninth win was just seconds away as the visitors wrapped up a tough match with another win, although the absence of Schweinsteiger was felt in the last hour.
Hannover 2, Schalke 2
Hannover, Germany - Teemu Pukki scored twice in his first Bundesliga start and Schalke tied Hannover, 2-2, on Sunday at AWD-Arena to extend its unbeaten run to three matches.
Pukki, a 21-year-old Finnish international, notched his first Bundesliga goal after 26 minutes, but an own goal later in the half and Mohammed Abdellaoue's eighth goal of the season in the second half put Hannover in front.
Schalke responded through Pukki in the 73rd when he turned and fired from just 8 yards out to the bottom corner.
Schalke lost two points to leader Bayern Munich with the draw, as it slipped from second to fifth and six points off the pace. Hannover is nine points off the pace in sixth.