The Colombia striker, 35, left Galatasaray having played just 25 minutes across three games in three competitions so far this season.
Falcao scored 20 goals in 43 matches in all for the Turkish giant, having signed as a free agent in 2019.
He now returns to LaLiga with Rayo, which was promoted through the play-offs last season.
Rayo registered its first win of the campaign ahead of the international break, beating Granada 4-0.
It will now be boosted by the experience of Falcao, whose previous stint in Spain saw him star for Atletico Madrid between 2011 and 2013.
Over two seasons, only Lionel Messi (96) and Cristiano Ronaldo (80) scored more LaLiga goals than Falcao's 52.
He averaged a goal every 112.4 minutes, scoring with 21.1 per cent of his 247 shots and marginally outperforming his expected goals value of 51.4.
Falcao, signed from Porto, went on to play for Monaco, Manchester United and Chelsea, ending his eight-year stay in Europe's top five leagues when he joined Galatasaray.
Between 2011 and 2019, his 122 goals in 212 games ranked him 14th across Europe.