The principality club has confirmed that it has secured the services of the Colombian on a five-year deal, seeing off competition from the Premier League
The Colombian has been in fine form for Diego Simeone's side, scoring 31 goals in all competitions as the capital club sealed Champions League football and won the Copa del Rey.
He had been widely linked with a move to the Premier League, with Manchester City and Chelsea vying for his signature, but has elected to continue his career in Ligue 1.
"Monaco are happy to announce that they have reached an agreement with Atletico Madrid and the player Radamel Falcao," a club statement read.
"Falcao, 27, has signed for five seasons. He will be presented to the media and start training with the group at a date to be confirmed by the club."
For his part, the Colombian said he was looking forward to the challenge at the club.
"I am very happy to join Monaco and very happy to wear these new colors," he said in a video message to fans.
Monaco president Dmitry Rybolovlev said that he was delighted that the former Porto man has chosen to represent the ambitious principality outfit.
"We are very happy that Falcao has chosen the challenge of Monaco for the future," he said. "It's a source of pride for us to have one of the best players in the world in the Monaco ranks."
Falcao joins former Porto teammates Joao Moutinho and James Rodriguez at Stade Louis II.