The deal for the 21-year-old Ghana international is reportedly worth £20m, but the Premier League champions have yet to disclose the actual figure.
Rahman told the club’s website: “I am very happy to sign for Chelsea, one of the biggest clubs in the world, and I can’t wait to train with my new team-mates and very much looking forward to working with them and Mr Mourinho.
“I am also looking forward to having the opportunity to meet the fans as soon as possible.”
Baba Raman Joins Chelsea from Augsburg. Image: Getty.
The full-back was a member of the Black Star squad which finished runners-up to Ivory Coast at the Africa Cup of Nations (Afcon) in February.
Owing to his prolific overlapping capabilities, he can also operate as a midfielder or winger.
He made more successful tackles than any other player in the Bundesliga last season and also attempted the most crosses from the left flank.
Raman joined Augsburg last summer and helped them qualify for the Europa League after making 31 league appearances.
He left Asante Kotoko to join then-newly promoted Bundesliga side Greuther Furth in 2012 and spent two years with the side.