The Spanish goalkeeper has signed a seven-year contract at Stamford Bridge.
Sanchez comes in as a replacement for Edouard Mendy, who has joined Al Ahli in Saudi Arabia.
Sanchez had been at Brighton since he was a teenager but lost his place towards the end of the 2022/2023 season to Jason Steele.
The 25-year-old made 23 appearances in the Premier League last season.
Co-sporting directors Paul Winstanley and Laurence Stewart said: ‘We’re very pleased to welcome Robert to Chelsea and he adds further quality to our goalkeeping unit.
‘Robert has repeatedly proved himself in the Premier League and been capped by his country. We are excited to watch him work with Mauricio and his coaching team during the season ahead.’
Sanchez's move to Chelsea sees him reunite with former Brighton goalkeeping coach Ben Roberts.
Sanchez has won two caps for the Spanish national team and was part of their squads for the 2021 European championships and 2022 FIFA World Cup in Qatar.
Chelsea have now signed seven players in the summer transfer window including Christopher Nkunku, Nicolas Jackson, Lesley Ugochukwu, Angelo Gabriel, Diego Moreira and Axel Disasi.
