Falcao, who did not feature for Monaco to start the Ligue 1 season due to an ankle injury, had been linked to Galatasaray throughout the transfer window.
Galatasaray confirmed negotiations with the striker this week before Falcao was welcomed by a crowd of thousands upon his arrival to Istanbul on Sunday night.
“An agreement has been reached with the professional football player Radamel Falcao Garcia Zarate and his club AS Monaco Football Club for a free transfer of the player," Galatasaray announced on Monday.
"According to the agreement, the player will be paid a net transfer fee of €5 million for three seasons starting from the 2019-2020 season."
He scored 83 goals during his stay with the club, but made it clear this summer that he was looking to move on from the team.
The 33-year-old striker had already embarked on two loan spells away from the club during his stint in Ligue 1, with the first being a spell with Manchester United in 2014-15 that produced just four goals in 29 appearances.
He didn't fare any better the next season as he was loaned to Chelsea, where he fired just once in 12 appearances for the Premier League club.
The forward did contribute 16 goals for Monaco last season, though, after scoring 24 and 30 the two seasons prior.
Falcao is Colombia's top all-time scorer, having fired 34 goals in 89 appearances while featuring at three Copas America as well as the 2018 World Cup.
The signing continues what has been a busy summer for Galatasaray, who are looking to build on a title-winning performance from last season.
Galatasaray have added a number of new faces this summer, including the likes of Stephen N'Zonzi, Emre Mor and Jean Micahel Seri on loan to go with moves for Ryan Babel, Jimmy Durmaz and Sener Ozbayrakli, who, like Falcao, have joined on free transfers.
Following the international break, Galatasaray will face Kasimpasa having collected four points from their opening three league matches.