Sorloth was on a two-year loan at Trabzonspor from Crystal Palace which would have ended after the 2020/2021 season.
Sorloth also popped up Tottenham’s radar after scoring 33 goals in 49 appearances across all competitions for Trabzonspor last season.
Leipzig will pay Crystal Palace £18m for Sorloth’s services.
The 24-year-old’s arrival will help fill the gap left by Timo Werner’s departure to Chelsea.
"After the initial talks, it quickly became clear to me that I wanted to play for RB Leipzig," said Sorloth.
"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."
Leipzig 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."
