The deal for the Colombia forward dragged on throughout Deadline Day and was finally made public long after 1am on Tuesday with the Old Trafford club having been granted an extension to complete paperwork.
Falcao was previously linked with moves to Arsenal and Manchester City, but it became apparent earlier on Monday that United were ready to dig deep to sign him to add to their attacking options.
His arrival at the club’s Aon Training Complex at Carrington was delayed and there was talk during the evening that he had failed a medical, although that proved to be completely groundless.
Falcao said: "I am delighted to be joining Manchester United on loan this season.
"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."
He later told MUTV: “It was a very long day. It was hard, difficult. We were waiting until the last minute but I didn't lose my faith. Now I want to say thank you to God for this. I'm very happy.
"For me, it was a dream to play in the Premier League. I always wanted to be here because this league is very good, it has very good teams and Manchester United is the best team in England. Everything is perfect for me.
"We have very top players. I admire them. I want to meet them for the first time. I am sure that this season will be great for us."
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."
Falcao has been plying his trade in Europe since Porto signed him for just over £3m from Argentinian side River Plate back in 2009.
After a prolific spell in Portugal, he joined Atletico Madrid in 2011, where more success in front of goal followed, and then secured a move to Monaco last summer for a fee in excess of £50m, but injuries badly affected his first season in France as he scored nine goals in 17 appearances.
He missed out on the World Cup with Colombia because of a serious knee injury suffered in January but scored two goals in his first three games this season with the French outfit.
The 28-year-old is the latest big-money arrival at Old Trafford this summer in the wake of van Gaal’s appointment as manager.
Deadline day also saw the arrival of Ajax player Daley Blind, who can operate in defence and in midfield and is well known to van Gaal from his time as Netherlands coach - he cost United around £14m.
Van Gaal finalised a British record £59.7m deal for Real Madrid midfielder Angel di Maria last week and the Argentina international made his debut in the 0-0 draw with Burnley on Saturday.
United’s summer business also included the signing of Southampton defender Luke Shaw, di Maria’s international colleague Marcos Rojo and Atletico Bilbao midfielder Ander Herrera.