He made 28 appearances in all competitions for Strasbourg this season. They finished seventh in Ligue 1 to book a spot in the Conference League play-offs.
Like Chelsea, Strasbourg are owned by American Todd Boehly and Clearlake Capital under the BlueCo umbrella, and Sarr's is the first transfer between the clubs.
The signing is Chelsea's fourth of the summer, following the arrival of Liam Delap for £30m from Ipswich Town, Estevao from Palmeiras for £29m and Dario Essugo from Sporting Lisbon for £18m.
BBC Sport understands that Chelsea are pursuing Borussia Dortmund winger Jamie Gittens.
Winger Jadon Sancho left Chelsea at the end of the season to return to Manchester United following his loan deal.