Szoboszlai has signed a five-year contract at Anfield to become Liverpool's second summer signing after Argentina World Cup winner Alexis Mac Allister.
The 22-year-old spent two seasons at the Red Bull Arena, scoring 20 goals and producing 21 assists in 91 appearances.
He has won the DFB-Pokal with Leipzig in each of the last two seasons.
Szoboszlai, who is already captain of his national team Hungary, faced Liverpool four seasons ago in the Champions League while at Austrian side RB Salzburg.
Liverpool manager Jurgen Klopp has fond memories of that teenager who is now taking over the iconic number eight jersey at Anfield.
"We already have a little knowledge of Dominik as an opponent because we had a couple of really tough games against RB Salzburg when he was a player there, and I think everyone who saw those fixtures will have realised that even at that stage he was a really interesting prospect," said Klopp.
"I'm pretty sure he was still a teenager at that time and since then he has made some big strides - moving to Leipzig, doing really, really well in the Bundesliga, playing for the Hungary national team and then becoming captain of his national team at an incredibly young age.
"This is a signing for our present and also for our future, and the work that has gone into it could not be more appreciated."
