Rangnick was expected to continue in the role of sporting director on a full-time basis after it was announced in 2018 that Julian Nagelsmann would replace him as first-team coach at the end of the 2018-19 season.
But the club has brought in the talented 38-year-old Krosche from Paderborn, while Rangnick will leave to become head of sport and soccer development at Red Bull.
The club confirmed the news in a statement, saying: "Markus Krosche is the club's new sporting director for the upcoming season. The 38-year-old arrives from newly-promoted Paderborn and has signed a contract until June 30, 2022.
"In addition, the two clubs have agreed a long-term partnership. The fee, at both clubs' behest, will remain undisclosed.
"Current sporting director Ralf Rangnick, at his own request, will become Head of Sport and Development Soccer at Red Bull from July 1. The 60-year-old will have an advisory role for football clubs in New York, Brazil and Leipzig."
Rangnick had been sporting director at Leipzig since June 2012, when he held the same post at both Leipzig and Red Bull Salzburg. He left the post at Salzburg to concentrate fully on Leipzig in May 2015 and achieved promotion to the Bundesliga before making way for Ralph Hasenhuttl to take over.
He returned to the helm in July 2018 and helped Leipzig to third place in the Bundesliga in 2018-19, 12 points behind champions Bayern Munich.
RB Leipzig CEO Oliver Mintzlaff said Rangnick had given his approval to the appointment of Krosche, who helped Paderborn rise from the third tier of German football to the Bundesliga in the space of just two years.
"We're pleased that we were able to secure Markus Krosche as our sporting director," Mintzlaff said.
"He's also the absolute preferred candidate of Ralf Rangnick, and with his appointment, we're getting a proven expert."