The Brazilian, who joins for a fee that could reportedly rise to 70 million euros, was unveiled before the Old Trafford crowd ahead of United's game against Liverpool.
"The opportunity to start a new challenge at Manchester United and in the Premier League is extremely exciting," Casemiro told the club website.
"I have worked with many great managers in my career and, having met with Erik (ten Hag) and heard his ideas, I can't wait to work closely with him, his staff and my new team mates in the coming years."
Casemiro had a hugely successful spell at Real after joining from Sao Paulo in 2013, winning five Champions League trophies, two league titles and one Copa del Rey, among other honours.
Capped 63 times by his country, Casemiro said a tearful goodbye to Real earlier on Monday, giving special thanks to fellow midfielders Luka Modric and Toni Kroos.