Second-half goals from the striking duo ensured die Roten picked up their fourth win in six Bundesliga encounters, while the home side slumps to 11th in Germany's top flight.
Mario Mandzukic capitalized on a poor Christian Molinaro back pass to round Sven Ulreich and open the scoring on the hour mark, before Thomas Muller doubled Bayern's advantage in the 70th minute, with Martin Harnik given his marching orders 10 minutes from time for the home side.
Bayern dominated possession early on, but took until the 10th minute to muster its first clear-cut opportunity after Dante rose highest to head Toni Kroos' corner over the Stuttgart bar.
Jupp Heynckes' men continued to press for the opener and again Kroos looked to turn provider from a free kick. His cross was behind Javi Martinez, who struggled to turn his header into the path of Ulreich, before David Alaba smashed his set piece from 30 yards out helplessly over the bar for the Bundesliga pace-setter.
The home team was struggling to contain Bayern, but mustered the best chance of the opening exchanges. Shinji Okazaki was played into the right and drove for goal on the half volley, only to flash his effort just wide of the far post on the half hour mark.
Stuttgart began to settle as the half drew on and came close to drawing first blood. Ibrahima Traore was picked out on the left during a swift counterattack and found Harnik unmarked at the far post. He headed back across goal, but Neuer gathered after almost putting the ball into his own net.
Bayern dominated possession still and after Stuttgart failed to clear from a Kroos corner, the ball eventually found its way to Mandzukic. The Croatian controlled well and went for goal with a half volley on the turn, firing over from 15 yards out with 10 minutes of the first half remaining.
Jupp Heynckes' side continued to control proceedings after the interval and was rewarded five minutes into the second half. Christian Molinaro played a weak back pass to Ulreich, which was capitalized on by Mandzukic. The Croatia international rounded the Schwaben No.1 and slotted into an empty net to give Bayern the lead.
The 26-year-old came close to adding a second just after the hour mark after being picked out by a Kroos free kick from the right. He headed downwards towards goal, but Ulreich dove well to turn his effort around the post.
Mandzukic turned provider with 20 minutes remaining after breaking down the right. He waited for the run of Muller in the 18-yard box before he picked out the Germany international, who beat Serdar Tasci to poke the ball past Ulreich from close range to double Bayern's advantage.
Stuttgart's evening went from bad to worse with 10 minutes to go after Harnik was given his marching orders by referee Florian Meyer for a second booking following a late tackle on Franck Ribery.
Bayern pressed for a third to round off a miserable encounter for the home side, but was made to settle for the 2-0 win as the club secured its 15th win of the season.