Visitors, Stuttgart took the lead through Martin Harnik after 25 minutes but Bayern hit back before half-time with goals from Thomas Muller, Toni Kroos and Luiz Gustavo.
Another three strikes in the space of six second-half minutes completed the rout, with Muller in fine form.
The 22-year-old scored his second between goals for both Mario Mandzukic and Bastian Schweinsteiger and the visitors' misery was compounded when Vedad Ibisevic was dismissed in the 74th minute.
Arguably Bayern's best goal came two minutes before the break, leaving Bruno Labbadia's side shell-shocked.
Gustavo drilled in a fantastic 35-yard effort that swerved away from Sven Ulreich and crashed in off the post.
In the day's other match, Hannover moved second behind leaders Eintracht Frankfurt with a 4-0 win at Wolfsburg.
Polish striker Artur Sobiech struck twice, in the 26th and 56th minutes, while Tunisian defender Karim Haggui (10th minute) and Danish midfielder Leon Andreasen (52nd minute) also scored.
All four goals were created by Hungarian midfielder Szabolcs Huszti.