The teenage forward makes the switch to Germany on a free transfer, having signed a contract until 2025.
"Alou Kuol has attracted attention as one of the top rookies in the Australian A-League and impressed us with his personality and ambition in our talks," said Stuttgart sporting director Sven Mislintat.
Kuol, who has scored seven goals in 16 games this season, commented:
"I am very happy and delighted to be given the opportunity to become a VfB Stuttgart player. VfB is a great club, where I want to take my next step.
"I will give everything and work hard to repay the trust of those in charge with my performances and develop as a player," stated the Sudanese-born striker.