The German has signed a six-month contract, which will then see him move into a consultancy role with the club for two years, subject to work visa requirements.
United had identified 63-year-old Rangnick as the man to see them through till the end of the current campaign following the dismissal of Ole Gunnar Solskjaer.
The respected coach leaves Russian side Lokomotiv Moscow to join the club, although Carrick remains in charge until his work visa is in place.
John Murtough, football director at the Premier League club, said: "Ralf is one of the most respected coaches and innovators in European football. He was our number one candidate for interim manager, reflecting the invaluable leadership and technical skills he will bring from almost four decades of experience in management and coaching.
"Everyone at the club is looking forward to working with him during the season ahead, and then for a further two years in his advisory role."
Meanwhile, Rangnick added: "I am excited to be joining Manchester United and focused on making this a successful season for the club.
"The squad is full of talent and has a great balance of youth and experience. 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.
"Beyond that, I look forward to supporting the club's longer-term goals on a consultancy basis."
Known by many back home as the godfather of modern German football, Rangnick has made his name taking relative minnows such as Hoffenheim and RB Leipzig from the lower divisions of Germany all the way to the Champions League.
Inspiring Jurgen Klopp and Thomas Tuchel along the way, Rangnick has impressed in managerial and director roles, building a global scouting network the envy of many a side.
Bringing through the likes of Erling Haaland, Sadio Mane and Roberto Firmino among many more, Rangnick has a revered reputation in Germany, boosted by two iconic managerial spells at Schalke.
Rangnick began working with Red Bull in 2012, seeing Leipzig and Salzburg assail to new heights, before he left to join Lokomotiv in July last year after a failed move back to Schalke as their director.
This won't strictly be the first time Rangnick has worked in England though, having previously played for West Sussex side Southwick during the 1979/80 season.
Rangnick has his work cut out for him at Old Trafford.
He joins a United side that has only won one of their last seven Premier League games, and sits 12 points off league-leaders Chelsea despite a summer of heavy investment.