The switch sees Marsch leave Red Bull Salzburg, affiliates of Leipzig, to rejoin the club he previously served as an assistant coach.
Leipzig said Marsch had agreed a two-year contract and would assume his new role on July 1.
Marsch has been a head coach before, most notably leading New York Red Bulls from 2015 to 2018, and the American should prove an easy fit with the German Bundesliga side.
Leipzig CEO Oliver Mintzlaff said: "In Jesse Marsch, we have been able to recruit our main head coach and quickly fill the most important position in the sporting side of the club with a top coach.
"Jesse has been part of the Red Bull football group for over six years now and has done incredible work in all his roles so far, constantly developing step by step.
"It's of course a great advantage that he already worked for a year at RB Leipzig. Jesse knows the club, the city of Leipzig and, above all, the club and playing philosophy.
"Alongside his qualities as a coach, Jesse is characterized above all by his positively ambitious style, which he uses to motivate and engage the people and around the club.
"We're looking forward to working with Jesse Marsch and to continuing our successful path with him."