"With Jesse Marsch, we have been able to hire our ideal candidate and fill the most important sporting position at the club with a top-level coach," said Leipzig sporting director Oliver Mintzlaff.
Marsch, 47, will take over on July 1 from current Leipzig coach Nagelsmann, who was announced as the next Bayern Munich trainer on Tuesday.
A former assistant coach of the US men's national team, Wisconsin-born Marsch has long been a rising star within Red Bull's sprawling global football network.
The Leipzig job would be his third in charge of a Red-Bull backed club, following a three-year spell at the helm of New York Red Bulls and two years in the Salzburg dugout.