The 21-year-old attacking midfielder has signed a contract with the Westphalian club until 2018 and will fill the void left by a serious knee injury to Jakub Blaszczykowski.
The Serbian passed his medical on Friday morning before joining his new team mates in training, although he is unlikely to feature in their clash with Eintracht Braunschweig on Friday night.
"This is a typical Borussia transfer," said the club's general manager Hans-Joachim Watzke to the Welt newspaper.
"It's one with a certain amount of risks, but also a big opportunity."
Dortmund Sporting Director Michael Zorc said on the club's official website: "We are delighted to have in Milos Jojic a versatile midfield player committed with great development potential, which completes our squad."