This last 16 German Cup tie was the reigning champions' first official game of 2009 after a one-and-a half month lay off, a training camp in the sunshine of Dubai, and five friendly wins.
They were quickly into their stride though with Bastian Schweinsteiger starting the rout in the 14th minute.
“We knew we were feeling good. We showed our quality,” said Schweinsteiger. “But we can play even better.”
Franck Ribery doubled up two minutes later and the France international then set up Luca Toni for Bayern's third just before the break. Schweinsteiger got the second leg of his double from the penalty spot in the 53rd minute with Brazilian Ze Roberto completing a fine night for the visitors before the hour mark.
It was Stuttgart's first loss since Markus Babbel took over as coach in November.
“That was an extremely bitter night for us,” said Stuttgart captain Thomas Hitzlsperger.
Bayern's league campaign resumes on Friday with a trip to Hamburg who earlier Tuesday won their Cup tie against 1860 Munich 3-1, while Schalke 04 saw off minnows Jena 4-1.
Bundesliga winter champions Hoffenheim were eliminated in the previous round by Freiburg, who were defeated 3-1 by fellow second-division outfit Mainz.
Wednesday's Cup programme features a showdown between top-flight sides Borussia Dortmund and Werder Bremen.