Tella departs the south-coast club for the German outfit after six years, having joined the Saints Academy from Premier League side Arsenal at the age of 17.
He scored three goals and registered six assists in 44 appearances across all competitions for Southampton before moving to Burnley on a season-long loan in August last year.
Tella helped the Clarets secure the Championship title, netting 19 times while providing five assists in 45 games.
The parties did not disclose financial details, but British media reported that the deal is worth around 20 million pounds.
"Although disappointed to lose such a talented homegrown player and popular character, the club felt the offer represented fair value for the 24-year-old," Southampton said in a statement.
"The move also gives him the opportunity to fulfil his desire to play abroad and play in the Europa League this season.
"We would like to thank Nathan for his contribution to the club, both in the Academy and first team, and for his professionalism throughout the summer transfer window, and wish him every success for the future."
