Filip Kostic gave the home side the lead on 15 minutes when he raced onto a hopeful long ball upfield that cut Eliaquim Mangala and Vincent Kompany open before beating Joe Hart.
Daniel Didavi doubled Stuttgart's advantage on the half-hour mark, capitalising on a slick one-two that left three City players for dead, and more lax defending from the visitors allowed Daniel Ginczek to score twice in four minutes as Manuel Pellegrini's men capitulated before half-time.
Raheem Sterling laid on goals for Kelechi Ihaenacho and Edin Dzeko in an improved second-half showing for City, but the damage was done.
