The Colombia international's move to Old Trafford was rubber-stamped several hours after Monday's night's deadline, with United agreeing a season-long loan with the option to buy.
Falcao is set to lead the line after being handed the historic No.9 shirt, which has been vacated since Dimitar Berbatov – now at Monaco – departed the club in 2012.
Other players to wear the iconic jersey in the Premier League era are Louis Saha, Andrew Cole and Brian McClair.
That quartet of United No. 9's scored 187 top-flight goals between them for the club, though only on one occasion – Berbatov in 2010-11 – did one of them top the Premier League scoring charts.
Falcao has worn the No. 9 throughout much of his career, including his time at the Stade Louis II, and is now likely to compete with existing forwards Wayne Rooney and Robin van Persie for a spot in the first team.
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The 28-year-old missed the World Cup in Brazil in due to a serious knee injury, with Teofilo Gutierrez wearing the No.9 shirt for Colombia and leading the line in his absence.
But Falcao has put his injury woes behind him to sign for United: “I am delighted to be joining Manchester United on loan this season,” he 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.”