The Bundesliga's top scorer took his tally to 15 goals and 23 in all competitions as Bayern extended their lead at the top.
Seventh-placed Stuttgart played two thirds of the game with 10 men after Cristian Molinaro was sent off although they still could have snatched a late equaliser.
Stuttgart went ahead in the sixth minute when Molinaro's cross was headed down by Martin Harnik for Christian Gentner to fire his shot past Manuel Neuer.
They enjoyed the lead for only seven minutes until Gomez got in front of his marker to turn Rafinha's cross from the right into the net.
Molinaro was given a second yellow card for a late tackle of Arjen Robben in the 29th minute and Gomez struck the winner when he turned in Philipp Lahm's pass just before the hour.
Bayern did not have it all their own way and Stuttgart nearly levelled when Cacau intercepted Daniel van Buyten's attempted headed pass to Neuer, but the Brazilian-born forward could not keep his shot down.
The visitors nearly snatched a win when Itay Shechter's header was stopped on the line by Weidenfeller and, although the Israeli forced over the rebound, his effort was disallowed for a foul on the goalkeeper.
