The 20-year-old had long been tipped to leave Salzburg following his exploits for the Austrian champions, with Napoli, Milan, Inter, Lazio and Arsenal all linked with a move for the Hungary international.
He has chosen to join Bundesliga side Leipzig, however, and will be available for their game against Borussia Dortmund after the mid-season break on January 9.
Szoboszlai has impressed for Salzburg this season, scoring four goals and setting up another six across 11 league appearances.
Szoboszlai, who has also scored three goals in 12 appearances for his country, told Leipzig's official website: "I'm very pleased that this transfer has been completed. The club has developed fantastically.
"It is among the best teams in the Bundesliga and is also doing well in the Champions League.
"This step is perfect for my development. I'm coming to Leipzig to work under a top coach and in an extremely ambitious team that is very young but also has a lot of experience and has been able to celebrate success.
"Here, I'll be able to continue my development incredibly and I want to help the club to continue its success in 2021."
Szoboszlai becomes the latest in a long line of talent to make the move between the two Red Bull-owned clubs.