Bayern coach Pep Guardiola dismissed talk of a crisis after the champions drew 1-1 against Schalke on Tuesday following a 4-1 hammering at Wolfsburg, and normal service was resumed by the Bavarian giants at the Mercedes Benz Arena.
The leaders were not at their best against Huub Stevens' strugglers, but Robben took his goal tally to eight goals in his last nine games and 14 for the season with a fine strike to open the scoring late in the first half.
Stuttgart have now lost their last 13 games against Bayern and have failed to score in their last four matches against any opponent, as they brace themselves for a battle to avoid the drop.
Stuttgart did not initially resemble a team facing a relegation battle and they went close to an opening goal when Sakai's measured strike clipped the outside of the far post following a swift counter-attack 27 minutes in.
Stuttgart had done a good job of keeping the Netherlands winger quiet until he burst ahead of Sakai to get on the end of Mitchell Weiser's pass before unleashing an unstoppable left-footed first-time strike that flashed past Sven Ulreich and found the far corner.
Muller replaced Weiser at half-time, while Stuttgart coach Stevens introduced Harnik and brought off Daniel Schwaab.
Stuttgart were unfortunate to be behind but things soon went from bad to worse for the home side when Alaba produced a moment of magic to double Bayern's lead six minutes into the second half.
The Austria left-back opened his Bundesliga account for the season in style, letting fly with a free-kick from 25 yards that gave Ulreich no chance.
