Bayern extends Bundesliga lead
The Croatia striker took advantage of a mistake from Christian Molinaro to put Bayern ahead in the 50th minute, intercepting the defender's back pass before rounding the goalkeeper.
The visitors put the game beyond doubt in the 72nd when Mandzukic crossed for Thomas Mueller to finish off a counterattack in the 72nd. Mueller earlier had a goal ruled out for offside.
It got worse for Stuttgart with 10 minutes to go when Martin Harnik earned his second yellow card for a foul on Franck Ribery.
Harnik came closest to scoring for the home side in the 35th, when his header struck the post and Manuel Neuer gratefully gathered the rebound.
Bayern's 15th win from 19 games leaves it 11 points clear of Bayer Leverkusen, which was held to a 0-0 draw at Freiburg on Saturday. Defending champion Borussia Dortmund is third, a point further back.