Following the end of this season, Rangnick will stay on at Old Trafford for an additional two years in a consultancy role.
His first game in charge of United, who are eighth in the Premier League, could be against Arsenal on Thursday if his work permit application is received in time.
If not, temporary manager Michael Carrick will remain in charge of the side.
“The squad is full of talent and has a great balance of youth and experience,” added Rangnick.
“All my efforts for the next six months will be on helping these players fulfil their potential, both individually and, most importantly, as a team.