Azpilicueta left Chelsea by mutual consent with one year remaining on his contract.
"Cesar leaves an everlasting mark at Chelsea, as a warrior, as a champion, and as loyal Chelsea legend," Todd Boehly, Chelsea chairman, and Behdad Eghbali, co-controlling owner, told the club's website.
"He has set the standards at the club for more than a decade and demonstrated to everyone what is required day in, day out to achieve success.
"Coaches and teammates past and present have trusted him to lead by example as captain. He has done so immaculately on and off the pitch, something for which we are grateful. For that and so much more, Cesar will always be welcomed back at Stamford Bridge."
Atletico head coach Diego Simeone had tried without success to sign the former Chelsea captain in previous transfer windows.
Azpilicueta was also targeted by Barcelona last summer but instead decided to extend his contract at Chelsea.
During 11 seasons at Stamford Bridge, he made 508 appearances, including 32 games last season. He won every possible trophy at Chelsea, including two Premier League titles and the Champions League.
Atletico also announced the signing of defender Alvaro Mourino on a permanent transfer from Racing Club.