Rangnick has signed a deal that will last until the end of the season but will stay on for two more years in a consultancy role when a permanent manager is appointed next summer.
The 63-year-old German will come in as a replacement for Ole Gunnar Solskjaer, who sacked last week after a poor run of form.
Manchester United have been managed by former player Michael Carrick in their last two games, winning away at Villarreal in the UEFA Champions League and drawing with Chelsea in the Premier League on Sunday.
Rangnick told United's website following his appointment: "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."
John Murtough, the football director at United, added: "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."
Rangnick could manage United for the first time against Arsenal on Thursday if his work permit issues are sorted on time.
