Burnic is a Germany Under-19 international defensive midfielder who made his Bundesliga debut in a 2-1 defeat against Darmstadt in February.
That was the only league game the 19-year-old featured in last season and Dortmund said they were eager to give him the chance to get more first-team experience at Bundesliga 2 champions Stuttgart.
The transfer was also helped by the fact that Stuttgart are managed by Hannes Wolf, who worked with Burnic until he left his role as Dortmund’s Under-19 coach in September last year to take up his current position.
"Dzenis has enjoyed great success with our new generation of players," Dortmund sporting director Michael Zorc said.
"We hope he gets as much game time as possible with Stuttgart in the coming season and we know he is in good hands with Hannes Wolf. We will be closely watching how he does."
Burnic made his Dortmund debut as a late substitute in their 2-1 UEFA Champions League win at Sporting Lisbon last October. He has been at the club since 2006 and won German championships at various youth levels in each of the past four seasons.
The teenager, who can also play in defence, said Stuttgart was a good place for a young player to continue their development.
"As a result, I'm really looking forward to being part of this team," Burnic said. "I'm going to give everything so that we can have a successful season."
He becomes the Swabians' fourth summer signing after they brought in attackers Anastasios Donis and Chadrac Akolo as well as midfielder Orel Mangala.