The 18-year-old will complete the move before the end of August, after undergoing a medical earlier this week, to become the first Chilean ever to play for the club.
Henriquez is one of the most sought-after players in South American football having notched up an impressive 10 goals in 15 league games, while he has also been prolific for Chile at Under-20 level.
|ANGELO HENRIQUEZ: EXPERT VIEW
Henriquez is mature well beyond his years, and whilst I think it would perhaps have been more beneficial for all concerned to leave him with La U for another year, I understand why Sir Alex wants him to begin the adaptation process as early as possible. Despite standing just below 6ft, he offers a serious aerial threat and keeps a cool head in front of goal. What is most impressive, however, has been his positioning and overall reading of the game. The teenager has taken the step-up to first team football impressively in his stride since the departure of Eduardo Vargas one year ago, fitting seamlessly into what is a very ‘Bielsan’ La U style.
"The player is physically and mentally prepared to take this determining step in his career."
President Jose Yuraszcek added: "Manchester United's representatives told us that it was fundamental for them to take him now."
Henriquez, who made his debut for the club earlier in the year, confirmed the move earlier in the week, describing it as "a dream".
"I am coming to stay at Manchester United and I'm going to play to obtain that objective," Henriquez told La Tercera.
"They are giving me this opportunity and I can't let it go, it is something that I have searched for my whole life. It's a dream I have always had and I can't afford to not want to be here.
"Everything is spectacular here in Manchester. For me it is all a dream. Imagine it, just to be able to play and share a dressing room with people like Wayne Rooney. It's a dream."
Henriquez is believed to have rejected an approach from Manchester City to sign for United.