The Bavarians wasted no time in getting out the blocks when Thiago Alcantara steered home the opener from a delightful pass by Joshua Kimmich on five minutes.
Stuttgart equalised at the midway stage of the stanza when Tossos Donis beat Manuel Neuer with a breathtaking long-range strike.
Donis threatened again after managing to get on the end of a high cross, but Neuer came up with a superb save to keep the hosts in the game.
After an eventless opening in the second half, Bayern regained advantage after Serge Gnabry finished off from poor defending after 55 minutes.
A minute later, Robert Lewandowski missed from close range, denying himself the opportunity to get on the score-sheet with an over the top finish.
Stuttgart nearly found the leveller after Nicolas Gonzales went one-on-one with Neuer, but the German staved off the effort with the former hitting the post on the rebound.
A minute later, Bayern received a penalty following a foul in the box by Gonzales on Lewandowski. The Polish international stepped up and missed the spot-kick with a poor execution.
Leon Goretzka showed up and made no mistake to finish off from close range after 72 minutes following a low pass to the box to extend Bayern's lead.
In the final minutes, Lewandowski redeemed himself after coming up with the fourth strike of the evening to give the hosts a resounding victory.