Sources have told ESPN the deal contains an option to buy next summer for a set fee.
The deal was thrown into doubt as the transfer had not been confirmed by either club before the midnight CET deadline.
Local media said the delay was caused by a problem with LaLiga's transfer registration system, however, and the documentation had been correctly submitted in time.
"We welcome Saul to the Club and believe he completes our squad as we compete for honours in five different competitions this season," Chelsea director Marina Granovskaia said.
"He is a proven winner, a player with huge experience and we know he will be very well-suited to the challenges that face us on all fronts in the coming year."
ESPN recently reported Chelsea were in talks with Saul, while Manchester United were also interested in the Spain midfielder, who has spent the last 13 years at Atletico after coming through the club's youth ranks.
However, United opted against pursuing a deal, leaving the path clear for Chelsea to make their second major signing of the summer after Romelu Lukaku joined from Inter Milan in a €115 million deal.
Information from Reuters was used in this story.