The 26-year-old flew into Manchester from Spain on Monday night and successfully underwent a medical before finalising a five-year deal on Tuesday.
The midfielder revealed United were the only club he was willing to leave Madrid for and said he cannot wait to work with Louis van Gaal.
"I am absolutely delighted to be joining Manchester United,” di Maria told the club's official website.
“I have thoroughly enjoyed my time in Spain and there were a lot of clubs interested in me, but United is the only club that I would have left Real Madrid for.
"Louis van Gaal is a fantastic coach with a proven track record of success and I am impressed by the vision and determination everyone has to get this club back to the top – where it belongs.
“I now just cannot wait to get started."
Van Gaal was delighted to finally secure the Argentina international’s services and said his pace and dribbling ability will add a new dimension to United’s attack.
"Angel is a world-class midfielder but most importantly he is a team player,” Van Gaal said. “There is no doubting his immense natural talent.
"He is a tremendously fast and incisive left-footed player who puts fear into the most accomplished defence.
“His dribbling skills and his ability to take on and beat opponents are a joy to watch. He is an excellent addition to the team."
Speaking after the 4-0 defeat to MK Dons, Van Gaal warned not to expect instant results from the former Real Madrid man.
"He has to adapt to our culture, our philosophy, he cannot change like that," Van Gaal said before clicking his fingers.
Van Gaal will experiment with Di Maria in several different positions over the next few days before deciding where he can slot the 26-year-old into his starting XI.
A change in system from 3-4-2-1 to 4-3-3 may be required, although the Dutchman thinks the player is versatile enough to play as a winger or a deep-lying central midfielder.
"You cannot switch to another system and it works like that," said Van Gaal, again clicking his fingers.
"You have to train it, it's a matter of time because the process needs time so we shall see if with di Maria we have to change the system or not."
The Argentine began his career at one of the most prolific youth academies in his homeland, Rosario Central.
In 2007, he moved to Europe to play for Benfica, before signing for Real Madrid in 2010.
The Argentina international has 52 caps for his national team, scoring 10 goals.