The issue of Colombia striker Falcao's future has been one of the stories of the latter part of the transfer window, with the former Atletico Madrid man linked with an array of clubs across Europe.
But United announced hours before the transfer deadline that they had struck an agreement to bring the 28-year-old to Old Trafford, though the deal was not confirmed until the early hours of Tuesday morning (GMT) after the club were granted an extension by the Premier League.
United are reported to have paid £6 million to seal a loan deal, which includes an option to buy Falcao for a rumoured €55 million (£43.5m).
"I am delighted to be joining Manchester United on loan this season," Falcao told United'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."
United manager Louis van Gaal added: "I am delighted Radamel has joined us on loan this season.
"He is one of the most prolific goalscorers in the game. His appearance-to-goal ratio speaks for itself and, when a player of this calibre becomes available, it is an opportunity not to be missed."
Juventus, Real Madrid, Manchester City and Liverpool have all been linked with the 28-year-old, with speculation mounting last Saturday as he was left out of Monaco’s 1-1 draw with Lille.
The same day reports of a message posted on the player's official Twitter account appeared to suggest that he was on his way to Real.
Falcao sought to clarify matters on Twitter, though, denying "a story about a tweet that I have not done."
And, after flying into Manchester by private jet on Monday, Falcao has become the latest high-profile signing to move to United following the captures of winger Angel di Maria from Real Madrid for a club-record free and Daley Blind from Ajax as new manager Louis van Gaal aims to restore the club to its former glories.
Falcao joined Monaco from Atletico Madrid for a reported £50million in May last year, but has struggled with injuries at the Ligue 1 outfit.
A serious knee problem, suffered in January, ruled him out of the World Cup and he has managed just 22 competitive appearances for Monaco, scoring 13 goals.
