But the 19-year-old will be loaned back to the Argentinian side for the 2024/25 season.
Anselmino has signed a seven-year contract at Stamford Bridge, having made 10 senior appearances for Boca, starting with his professional debut in June 2023.
Anselmino, who scored his first career goal in a Copa Sudamericana win over Trinidense in April, is Chelsea's eighth signing of the summer transfer window.
London club Chelsea, who finished sixth in the Premier League last season, begin the new campaign at home to champions Manchester City on August 18.
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