The Colombia international, who had been linked with a host of clubs this summer including Real Madrid and Arsenal, will spend the 2014-15 campaign on loan at Old Trafford.
As reported earlier in the day, the Red Devils will pay a 10 million pound-plus loan fee for the 28-year-old in addition to a significant portion of his £300,000-a-week Monaco wages and will have the option to buy at the end of the loan.
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Falcao flew into Manchester by private jet on Monday afternoon to complete a medical at the club's Carrington training base and pen personal terms.
The Colombia international becomes United's sixth summer signing after Luke Shaw, Ander Herrera, Marcos Rojo, Angel Di Maria and Daley Blind.
"I am delighted to be joining Manchester United on loan this season," Falcao told the club's official website.
"Manchester United is the biggest club in the world and is clearly determined to get back to the top. I am looking forward to working with Louis van Gaal and contributing to the team’s success at this very exciting period in the Club’s history."
The former Atletico Madrid forward's arrival allowed United to ship out both Javier Hernandez on loan to Real Madrid and Danny Welbeck to Arsenal during a busy deadline day.
Falcao only moved to Monaco from Atleti last summer, signing a five-year deal at the time, but becomes the latest player to join the principality club's exodus, with compatriot James Rodriguez also leaving the Stade Louis II.