The 26-year-old has signed a four-year contract with the German side.
Audel joins Stuttgart for an undisclosed fee and becomes the Bundesliga side’s fifth signing of the summer.
Stuttgart have also signed Martin Harnik, Philipp Degen, Christian Gentnar and Marc Ziegler this summer.
“Johan fits in perfectly at Stuttgart with his specific qualities. We’re convinced that he will quickly find his way at the club and in the Bundesliga,” Stuttgart’s technical director Fredi Bobic was quoted as saying on the club’s official website.