The Chile youth international has moved to Old Trafford from Universidad de Chile for an undisclosed fee, reported to be in the region of £3.5 million.
United had to wait until the striker obtained a visa and work permit before he could travel to Manchester and complete his transfer to the club.
Sir Alex Ferguson said: “Angelo has made a tremendous impact on the game in his short career so far. 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.”
United have had an option to purchase Henriquez ever since he impressed against the club's youth team at the age of just 14.
The striker said: “I am thrilled to be joining the biggest club in the world. 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. I hope I can be part of many successful years with United.”
Henriquez has scored 14 goals in only nine games for Chile's Under-20 side.