Szoboszlai, 20, has signed a four-and-a-half year contract with the club that will keep him at the Red Bull Arena until June 2025.
"With the signing of Dominik Szoboszlai, we've been able to further strengthen our squad," Leipzig CEO Oliver Mintzlaff said to the club's website. "With Dominik Szoboszlai, the squad has depth in more areas and gets more quality."
The Hungary international spoke of his desire to work with coach Julian Nagelsmann, while also praising the club for its development over the last few years.
"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," he said.
"Here, I'll be able to continue my development incredibly, and I want to help the club to continue its success in 2021.
"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."
Szoboszlai has been in fine form this season, scoring eight goals and assisting 10 in 20 matches. In the previous campaign, he scored 12 goals and set up a further 18 in 40 matches for Salzburg.
Szoboszlai will link up with his Leipzig teammates on Jan. 9 and is free to make his debut in the match against Borussia Dortmund on the same day.
Szoboszlai will wear the No.17 shirt at Leipzig.