Goals from Claudio Pizarro and Thiago Alcantara secured a 16th win of the season for the league leaders but they only clicked into gear in the final 15 minutes of the game on Wednesday.
Pizarro headed Bayern's equaliser in the 76th minute and a sensational acrobatic effort from Thiago sealed all three points in the last minute of stoppage time.
Stuttgart took the lead just in the 29th minute through Vedad Ibisevic, the striker reacting quickest to fire past Manuel Neuer.
Bayern's win extends their lead at the top of the table, and ensures their 43-game unbeaten run in the league continues.
Bayern named an unchanged side for the first time in Pep Guardiola's reign, with rumoured Juventus target Mario Mandzukic named as a substitute.
Bayern survived a penalty shout against Rafinha four minutes later, Werner's header struck the right-back's arm inside the area, however referee Manuel Grafe was not convinced and waved home appeals away.
Guardiola's side were far from their normal fluent self and were punished in the 29th minute as Ibisevic netted his 10th league goal of the season.
Werner's deflected shot from the edge of the area dropped to the Bosnian striker via the boot of Abdellaoue, who looked offside, and he calmly fired past Neuer from 10 yards to give the hosts a deserved lead.
Ibisevic had a chance to double his tally, and Stuttgart's lead, early in the second half but he fired wide of goal after a mistake from Rafinha.
Fellow substitute Mandzukic had a chance to win the game in the closing minutes but Thiago spared Bayern's blushes with an acrobatic volley in the last seconds of the match.
