Henriquez, 18, has joined from Universidad de Chile for an undisclosed fee. British media said the player underwent a medical and was granted a work permit last month.
"Angelo has made a tremendous impact on the game in his short career so far," United manager Sir Alex Ferguson told manutd.com.
"His pace is a great asset and he reads the game really well for someone of such a young age.
"We have a proud record of developing young talent, and Angelo fits the profile of a United player - skilful, attacking and someone who plays the game with real flair."
He also told Inside United magazine: "We scouted him when he was 14-years-old and just monitored him. When he turned 18, we knew we could take him; it's the first time we can sign them in this country, at 18."
Henriquez said: "Manchester United is a great team and has a great manager. To be able to be part of that is a huge honour, and I can't wait to get started."
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