PSG host relegation battling Le Havre on Saturday
The 29-year-old spent last season on loan at Manchester United, scoring four goals in 29 games.
"I am very happy to be joining Chelsea and can't wait to start training and help with our aim of retaining the league title and being successful in Europe," Falcao told the club website.
Falcao previously played for River Plate, Porto and Atletico Madrid.