The Red Devils announced the appointment of the former Hoffenheim, Schalke and RB Leipzig boss on Monday, eight days after the departure of Ole Gunnar Solskjaer.
Rangnick, 63, will take charge of the team for the rest of the season before taking up a two-year consultancy role at Old Trafford. He leaves a post as head of sports and development at Lokomotiv Moscow.
Murtough 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."
Rangnick, who is known for his high-pressing philosophy and influence on some of Bundesliga's brightest minds, started his career with his hometown club Viktoria Backnang.
After spells with the likes of Stuttgart, Hoffenheim, Hannover and Schalke, Rangnick enjoyed a brief managerial stint with RB Leipzig, who rose from the fourth tier of German football to the Bundesliga with the new United man as director of football.
Across 293 games in the German top flight, Rangnick has 120 victories to his name at a win percentage of 41.
"I am excited to be joining Manchester United and focused on making this a successful season for the club," Rangnick said.
"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."