Both sides went into the international break winless in their previous three matches, so perhaps it wasn’t a surprise to see the first half plagued by a lack of attacking quality despite the superstar names in the home ranks.
Bayern were quick out of the blocks, but Serge Gnabry curled wide inside three minutes when he really ought to have done better, while Raphaël Guerreiro struck a tame effort straight at Alexander Nübel.
Chances were hard to come by from there for the Rekordmeister, who lost young midfielder Aleksandar Pavlović to injury after an awkward landing following an aerial duel.
England captain Harry Kane missed several half-chances for the hosts, who were left frustrated to be heading into the interval without the lead considering Stuttgart had offered very little to threaten them at the other end.
Vincent Kompany’s men continued to probe after the restart, with Kane left stunned that he was unable to convert Gnabry’s square cross from close range.
That miss almost came back to haunt the striker when Stuttgart finally created a chance, though Josha Vagnonman fluffed his lines when firing over from Jamie Leweling’s cutback.
Having survived that scare, Bayern and Kane would not be denied again as, after being left in far too much space, the Englishman arrowed a low strike into the bottom corner from long range.
That ended a four-match scoring drought for Kane for club and country, but just four minutes later he doubled his tally to put the contest beyond the visitors.
Gnabry’s cross into the box wreaked havoc among the Swabians’ backline, and after bringing the ball under his spell with a delicate flick, Kane slammed home what was already his 43rd Bundesliga goal in just his 39th game - no player in the division’s history has reached that tally quicker.
And, after squandering a big chance to make it a hat-trick, Kane pounced in the 80th minute, seizing on a loose ball after João Palhinha’s shot was blocked and prodding home from close range.
Kingsley Coman added a fourth late on to round off a successful night for Kompany’s side, who inflicted a fourth away defeat of the campaign on Stuttgart, whose winless streak extended to four matches overall.
Flashscore Man of the Match: Harry Kane (Bayern Munich)