With UEFA Champions League qualification secure, a victory for Stuttgart would see them move within two points of Der Rekordmeister in second.
And the hosts got off to the perfect start when Leonidas Stergiou broke the deadlock midway through the first half, dinking a first-time finish over an onrushing Manuel Neuer.
However, the jovial mood at the MHP Arena darkened slightly when referee Tobias Welz pointed to the spot after Bayern man Serge Gnabry was brought down in the penalty area by Waldemar Anton.
Despite a lengthy VAR check, Harry Kane kept his cool from 12 yards, stroking home in typical fashion to level things up.
Kane remained influential in the second half, this time as a creator, with the Englishman’s pin-point pass playing Eric Maxim Choupo-Moting through on goal. Unfortunately for Bayern, the veteran striker lacked the composure to convert, dragging his strike wide of the post.
At the opposite end, Deniz Undav went close, with his drilled effort saved by Neuer.
The Bayern shot-stopper’s resilience didn’t deter Stuttgart from pushing for a winner, which eventually came seven minutes from time. Arriving late at the back post, Jeong was picked out by fellow substitute Silas’ cross.
The provider would soon turn goalscorer, when Silas capped off a deserved victory for the UCL qualifiers deep into added time.
The triumph means Sebastian Hoeneß’s men have now lost just one of their previous 20 home matches, while Bayern will now have to pick themselves up ahead of Wednesday’s UCL semi-final second leg against Real Madrid at the Santiago Bernabéu.
Flashscore Man of the Match: Jeong Woo-yeong (VfB Stuttgart)