Hosts Schalke took a third-minute lead through Joel Matip's close-range effort and Kyriakos Papadopoulos rose above two defenders at the near post to drill home a header in the 57th.
Teenager Julian Draxler finished off a quick passing move involving Klaas-Jan Huntelaar and Raul to grab the third goal 10 minutes from time before lacklustre Stuttgart notched a late consolation through Shinji Okazaki.
Bayern are top on goal difference despite slumping to a 3-1 defeat at Borussia Monchengladbach on Friday, the first game since the league's winter break.
"I was surprised by how strongly we started," Schalke coach Huub Stevens told reporters. "Tactically we played a very strong game. We now have as many points as Bayern and it is nice to dream."
HEAD INJURY
The only sour note for Schalke was the head injury sustained by captain Benedikt Howedes.
He could be out for some time after colliding with team-mate Marko Hoeger just before the break and being taken to hospital.
VfL Wolfsburg coach Felix Magath fielded four of his eight new signings in the 1-0 victory over Cologne but it was youth academy product Sebastian Polter who headed in the only goal from a Christian Trasch cross.
Werder Bremen missed the chance to close in on the top spots, drawing 0-0 at lowly Kaiserslautern to remain in fifth place on 30 points after 18 games.
Nuremberg kicked off their year in style, beating fellow strugglers Hertha Berlin 2-0 to move above their opponents into 12th spot with 21 points.
In another relegation battle Freiburg struck two minutes from time through Matthias Ginter to snatch a 1-0 win over bottom club Augsburg while Hoffenheim and Hannover 96 shared a 0-0 draw.
