Chelsea will paid £3.4 million (€4 million) for Saul, with the option to make the signing permanent for £30 million (€35 million) next summer.
After breaking into the first squad as a 17-year-old in 2012, the 26-year-old made more than 300 appearances for Diego Simeone's title-winning side last season.
Saul told Chelsea's website: "I am very excited to start this new challenge with Chelsea, "Fans of the Blues, I am now one of you, and I can't wait to put on the shirt, begin training, and meet you all. See you soon!"
Saul's transfer was confirmed after the Premier League transfer window closed at 11 p.m. on Tuesday night, but it was allowed to proceed because the paperwork was submitted on time.
Following the acquisition of third-choice goalkeeper Marcus Bettinelli from Fulham and the club-record purchase of Inter Milan striker Romelu Lukaku, he is Chelsea's third and final summer signing.
Thoughts on Chelsea's transfer window?
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