Financial details of the deal were not disclosed, but British media reported that Chelsea signed Sarr in a deal worth around 14 million euros ($16 million).
The 19-year-old centre back, son of former Senegal international Pape Sarr, made 28 appearances for Strasbourg after joining them from Olympique Lyonnais in August. He has represented France at Under-19 level.
Strasbourg and Chelsea are part of the same multi-club ownership structure, with the French side being taken over by Todd Boehly's consortium BlueCo in 2023.
Chelsea begin their Club World Cup campaign against Los Angeles FC on June 16.