Dutt was sacked as coach of Bremen in October, but negotiations continued with regards to his contract, which was due to expire in June 2016.
However, Bremen confirmed the mutual termination of the deal on New Year's Day, with Dutt tipped to take over from Fredi Bobic at Stuttgart.
That move has now been fianlised, with Dutt signing a deal until December 2018.
"We had clear specifications for this position as director of sport," said club president Bernd Wahler. "The important points, aside from sporting competence, were the implementation of our strategy and the patency from the youth teams right through to the professional first team.
"Further factors included leadership qualities, team and communication abilities, as well as an identification with our values.
"His economic understanding is also important in dealing with our budget and processes.
"Having weighed up all aspects, I am convinced that Robin Dutt is the correct solution for the role of Director of Sport at VfB."
Stuttgart sit 15th in the Bundesliga and Dutt believes the immediate target should be top-flight survival.
"In the constantly changing surroundings of the Bundesliga, the club finds itself facing a major challenge," he said. "In the short-term, we will be doing everything in our power to ensure we are successful in the fight for top-flight survival.
"In the mid to long-term, we intend to combine the different departments and responsibilities in the sporting areas, in order to ensure we can initialise positive and sustainable developments for the future."
Dutt parted company with Bremen after nine matches of the season, with the club then bottom of the table and without a win.
