Bayern was frustrated for much of the first half at Mercedes-Benz Arena, but former Schalke prodigy Goretzka made the breakthrough shortly before the interval, curling home at the culmination of a fine team move.
Goretzka's goal punctured Stuttgart's spirit, and although the woodwork denied David Alaba, Robert Lewandowski thumped home in the 62nd minute to double the advantage.
Bayern was not finished there, though, Thomas Muller getting in on the act late on to cap off the win in style.
Stuttgart had looked comfortable early on despite a typically dominant start from Bayern — Ron-Robert Zieler only called into action once in the opening stages as he kept out Joshua Kimmich's 16th-minute volley at the near post.
But Stuttgart's resolve was broken in the 37th minute, Goretzka finding enough space on the edge of Stuttgart's area to curl in a sublime finish.
Bayern clicked up a gear, and only an instinctive stop from Zieler prevented Mats Hummels extending its lead prior to break, before Alaba rattled the upright with a wicked free-kick after the restart.
Any lingering Stuttgart hopes were ended just after the hour, Goretzka turning provider with a neat pass into Lewandowski, who unleashed a rasping strike into the bottom-right corner.
Bayern added further gloss to the victory with 14 minutes remaining, Lewandowski and Muller combining superbly for the latter to tuck home.
Bayern plays host to pointless Bayer Leverkusen after the international break, while Stuttgart travels to the Black Forest to take on Freiburg.