Rangnick will be interim manager until the end of the season before the former Schalke 04 and RB Leipzig coach takes up a consultancy role with the club for a further two years.
Rangnick, who began coaching in the early 1980s, is a proponent of ‘gegenpressing’ — a style of football in which teams press with high energy to win the ball back as soon as possible, rather than falling back to regroup.
He left his role as head of sport and development at Russian Premier League side Lokomotiv Moscow to take over at United.
Under caretaker Carrick, United beat Villarreal 2-0 in the Champions League and drew 1-1 with Chelsea in the Premier League last week. They are 10th in the standings on 18 points after 13 matches, six points off the top four with a game in hand. Rangnick’s first game in charge will be Sunday’s league game at home to mid-table Crystal Palace.