The visitors grabbed the lead when Stuttgart defender Arthur Boka's pass found Bastian Schweinsteiger outside the box and he unleashed a ferocious drive past helpless Stuttgart keeper Jens Lehmann.
Two minutes later French playmaker Franck Ribery combined well with Ze Roberto and Luca Toni before the Italian found Ribery for an easy tap-in.
Bayern, cracking a fragile Stuttgart defence at will, could have added a third but Ribery's weak penalty, awarded for a handball, fell into Lehmann's arms.
Ribery quickly made amends, setting up Toni with a clever pass two minutes from the break which allowed the Italian to curl a shot past the Stuttgart keeper.
Schweinsteiger claimed his second 10 minutes into the second half with a penalty before Ze Roberto added a fifth.
Stuttgart's Mario Gomez scored a late consolation goal.
"The boys quickly understood Stuttgart were waiting for their chance and not coming forward so we could attack and score," Bayern coach Juergen Klinsmann told reporters.
Stuttgart coach Markus Babbel said his team was too scared.
"The way we lost was unacceptable," the former Bayern player said. "I can accept defeat but not like this. We were too scared. I have to admit though Bayern were outstanding tonight."
OLIC HAT-TRICK
Hamburg SV striker Ivica Olic scored a hat-trick against 1860 Munich to lead his team into the quarter-finals and show why Bayern Munich rushed to sign him for next season.
Olic broke the deadlock a minute before halftime after 1860 Munich keeper Philipp Tschauner's poor pass was intercepted by Piotr Trochowski. Olic added two more before Munich pulled one back with Daniel Bierofka in the 86th minute.
Schalke also progressed with a 4-1 win over Carl Zeiss Jena following a brace from Jefferson Farfan. The Peruvian striker tapped in a low cross from Ivan Rakitic after 19 minutes and added a second in the 65th.
Jena pulled one back through Andre Schembri before Rakitic and Kevin Kuranyi sealed Schalke's win.
Schalke had Gerald Asamoah sent off for a foul on Rene Eckardt while the hosts also had Marco Riemer dismissed for a foul on Kevin Kuranyi.
Mainz beat Freiburg 3-1 in the day's other match.
Four more matches are scheduled for Wednesday.
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
