The 20-year-old Hungary attacker reportedly cost 20 million euros ($24 million) and has signed a contract until June 2025 in a move from the teams owned by energy drink brand Red Bull.
Szoboszlai had also attracted the attention of Bayern Munich and Real Madrid by scoring 26 goals, and creating 34 more, in 82 games since joining Salzburg in July 2017 from their second division feeder club FC Liefering.
In August, Szoboszlai told German magazine Kicker he wanted to play in one of Europe's best leagues with "a top five team".
After Salzburg failed to reach the last 16 of the Champions League, Szoboszlai could line up against Liverpool in Leipzig's home first leg on February 16.
Leipzig's chief executive Oliver Mintzlaff said they invested the money from reaching the knockout stages to sign Szoboszlai and bring "additional quality" to their squad.
The transfer window only opens next month, so Szoboszlai is only eligible to make his debut against Borussia Dortmund on January 9, which could pit him against his ex-Salzburg team-mate Erling Braut Haaland.
The pair spent 12 months together at Salzburg in 2019, when Haaland scored eight goals in six Champions League group stage games with Szoboszlai on the left wing.
Having knocked Manchester United out of the Champions League with a 3-2 home win in their final group stage game to reach the knock-out stages, Leipzig are deep in the Bundesliga title race.
They earned a 3-3 draw at defending champions Bayern Munich a fortnight ago.
Their 1-0 win at Hoffenheim on Wednesday leaves Leipzig third in the German league table, level on 27 points with Bayern and one behind leaders Bayer Leverkusen.