The 18-year-old has been on United's radar all summer after an impressive year for the Chilean giants in which he scored 11 goals in 17 games.
"I am thrilled to be joining the biggest club in the world. Manchester United is a great team and has a great manager," said Henriquez on United's official website.
"To be able to be part of that is a huge honour and I can't wait to get started. I hope I can be part of many successful years with United."
United manager Sir Alex Ferguson too expressed his delight at signing the 18-year-old.
"Angelo has made a tremendous impact on the game in his short career so far," said the Scot. "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."
Henriquez, a Chile Under-23 international, will wear the number 21 shirt for United in the 2012-13 season.