Beach arrives in west London to join up with the Academy side for the 2022/23 season from Southampton down on the south coast.
The 18-year-old and Wales international won the U18s Premier League South title with the Saints last season, before falling to defeat to Manchester City in the National final.
He won’t be playing for Thomas Tuchel’s side any time soon, but the signing has left Jim Fraser, Chelsea's assistant head of youth development and head of youth recruitment, delighted with the acquisition.
Fraser told the official Chelsea website: “We are very pleased to bring Eddie to the club.
“He is a talented goalkeeper with good potential and will join our development squad for the coming season.”
Chelsea are believed to have beaten a number of clubs to the highly talented youngster after agreeing a fee with Southampton last week.
Posh boss Grant McCann was pleased to strike an agreement with Chelsea, saying: "Lucas is very highly-rated at Chelsea.
"He is commanding, a very good shot-stopper and extremely aggressive from crosses and good with his feet.
"I must give huge thanks to Carlo Cudicini and Chelsea for trusting us in developing Lucas further and we can’t wait to work with him."
Chelsea's Academy is set to be fully supported under the Club's new ownership.
As stated by Nizaar Kinsella: “Neil Bath and the Academy will be massively backed by Todd Boehly. Boehly has already been at training games that the Academy has been playing.
"They’ve been signing all the targets that Chelsea wanted.”
Boehly, co-controlling owner, Chairman and interim sporting director of Chelsea, confirmed his intentions for the Blues following the confirmation of the board's restructure, which revealed Marina Granovskaia's departure.
He said: "As custodians of Chelsea FC, we now begin executing our long-term vision and plan for the Club, creating an outstanding experience for its passionate, loyal fans, and continuing to challenge for top honours in line with Chelsea FC's decorated history.
"Working together, side-by-side, we are firmly committed to winning, both on and off the pitch. For us, that effort has begun."