Friday's win was a perfect morale booster ahead of next week's Champions League date with Bayer Leverkusen.
Michael Olise in the 45th, Leon Goretzka in the 64th and substitute Kingsley Coman in the 90th overturned Stuttgart's lead from Angelo Stiller in the 34th.
Especially Bayern's second and third were gifts of sorts as Stuttgart couldn't follow up on a strong first half.
Stuttgart goalkeeper Alexander Nübel first played the ball to Stiller whose pass attempt was intercepted by the lurking Goretzka, and Coman stole the ball from Josha Vagnoman and hit the empty net from a distance.
"It was a lot of work. We didn't get into the game well but we stayed calm. We did better in the second half and got a deserved victory," Goretzka told streaming portal DAZN.
Bayern increased their big lead over title holders Leverkusen, who can reduce it to eight points again if they win at third-placed Eintracht Frankfurt on Saturday.
Bayern host Leverkusen on Wednesday in the first leg of the Champions League last 16 tie, with the return leg six days later.
Last season's runners-up Stuttgart remain seventh. They have garnered only four points from the last six games and lost at home for the third time in a row.
"We make it very easy for our opponents to score. It shouldn't have happened," Stiller said, and Nübel admitted his pass was "too risky" and "that we made too many mistakes."
Bayern had striker Harry Kane and left back Alphonso Davies back in the starting 11 from injuries but Joshua Kimmich and Aleksander Pavlovic were missing, and it was Stuttgart who threatened first.
A long ball bounced between two Bayern players into the path of the unmarked Vagnoman who however only managed a tame effort into the arms of goalkeeper Manuel Neuer.
At the other end, Jamal Musiala aimed wide from a left angle after being set up by Leroy Sane.
Deniz Undav should have then opened the scoring for the hosts but the goal eventually came in the 34th when Stiller smashed into the roof of the net from the edge of the area.
Stuttgart beat Bayern 3-1 at home last season but Bayern were level on the stroke of half-time when Sane's defence-splitting pass was converted by Olise into the far left corner - and then mainly controlled the action after the break.
Kane's shot in the 64th was too harmless to beat Nübel, but the lead came a minute lead later when Goretzka pounced to steal the ball from Stiller and fire home.
Vagnoman then lost the ball in midfield to Coman who ran past Nübel who was far out of his penalty area and converted from the right into the empty net.
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