RB Leipzig signed Alexander Sorloth from Crystal Palace in a deal worth €22million (£20.5m).
Bundesliga club Leipzig agreed to pay Trabzonspor €10m to terminate Sorloth's loan from Palace, which was due to run until the end of the 2020-2021 season.
The Premier League side will also receive €10m (£9.3m) up front, with a further €2m (£1.9m) payable in add-ons to be split between Palace and Trabzonspor.
Norway international Sorloth, 24, has put pen to paper on a five-year contract at the Red Bull Arena after scoring 24 goals in 34 Super Lig appearances last season.
Despite the arrival of Hwang Hee-chang from Salzburg, Julian Nagelsmann was still keen to bolster his attacking options after seeing Timo Werner leave for Chelsea and Patrik Schick join Bayer Leverkusen after his loan from Roma ended.
"After the initial talks, it quickly became clear to me that I wanted to play for RB Leipzig," Sorloth said. "The club were impressive in the league last season, and even more so in the Champions League.
"The attacking brand of football that Julian Nagelsmann plays suits me well. I'm looking forward to challenge together in the Bundesliga as well as the Champions League."
CEO Oliver Mintzlaff added: "We're very pleased with the signing of Alexander Sorloth. Successfully pulling off this complex transfer was only made possible thanks to the good co-operation among our team, who I'd expressly like to thank.
"Not only does Alexander fit our philosophy perfectly, he is a tough centre forward who can also play out wide, who is the missing piece to our squad-planning puzzle this season."
Prior to 2019-2020, Sorloth's best return in a league season was 13 goals from 26 appearances for Bodo/Glimt in the 2015 Eliteserien.
Spells with Groningen, Midtjylland, Palace and Gent preceded his move to Trabzonspor last year.