The hosts fell behind at the Allianz Arena to a 25th-minute volley from Martin Harnik.
However, two goals in quick succession just after the half-hour mark swung the game in Bayern's favour.
Luiz Gustavo then scored from 30 yards out to give the Bavarian giants a commanding 3-1 lead at half time.
Mario Mandzukic got in on the act in the second half, before Muller doubled his tally for the afternoon in the 49th minute.
The rout was completed by Bastian Schweinsteiger in the 51st minute when he headed home from Muller's cross.
Stuttgart were reduced to 10 men when Vedad Ibisevic was shown a red card for a headbutt on Jerome Boateng.
Javi Martinez came on to make his debut in the 77th minute.