Leipzig parted ways with Ralph Hasenhuettl after a disappointing 2017/18 season which saw them finishing sixth in the German top flight and miss out on Champions League qualification.
The 60-year-old Rangnick was previously coach of the club as they made their way through the divisions between 2012 and 2016, before retiring and becoming sporting director.
But with Julian Nagelsmann due to join in July 2019, from Hoffenheim, Rangnick has stepped in as a stop-gap.
Rangnick said that Leipzig -- who finished second to Bayern Munich two seasons ago in their debut Bundesliga campaign -- were hoping to bring Everton forward Ademola Lookman to Germany permanently after a successful loan spell.
"We're still in talks over Ademola Lookman -- we want him back and he wants to come," he said.