Dortmund have acquired Adeyemi's services for a reported €30m and he has signed a five-year contract at the Signal Iduna Park.
The 20-year-old will come in as a replacement for Haaland, who will join Manchester City at the end of the season.
Adeyemi will be the third summer signing for Dortmund, who will also welcome Niklas Sule and Nico Schlotterbeck from Bayern Munich and Freiburg respectively this summer.
"After the transfers of Sule and Schlotterbeck, we are getting another very exciting player for next season with Karim Adeyemi," Sebastian Kehl, who will take over as sporting director from 2022-23, told the club's website.
"What is more, he is someone who has a strong affinity for BVB, who supported the Black and Yellows as a youngster and who decided to sign for Borussia Dortmund amidst several offers from Europe's top leagues.
"His speed and goal-scoring ability are impressive, and although Karim, at just 20 years old, is already at a very advanced stage in his development as a footballer and athlete, we still see enormous potential for more."
Adeyemi is the Austria league's top scorer this season with 19 goals in 27 appearances. He also struck four times in 10 Champions League appearances for Salzburg, who reached the last 16 before getting knocked out by Bayern Munich.
"As a young boy I was fascinated by the Black and Yellows' fast-paced football," Adeyemi said. "That's why, as soon as I heard Dortmund were interested in me, I knew I wanted to sign with BVB.
"I made the conscious decision to sign a long-term deal because I'm convinced we're going to form an exciting team that, with the support of the amazing fans in Dortmund, will be able to compete for and win titles in the years to come."
Adeyemi has won three caps for the German national team, scoring one goal.
