Sarr, 21, left French club Nice this summer and has signed a five-year contract at Stamford Bridge.
The Premier League also announced that the defender will spend the 2020-21 campaign out on loan.
"The opportunity to sign Malang was one we could not miss," Chelsea director Marina Granovskaia told the club's website.
"He is a tremendous prospect and we will be monitoring him closely during his loan period, hoping he will soon be back at Chelsea. We are very happy to welcome him to the club today."
Sarr made 116 appearances for Nice and represented France at last year's under-21 European championship.