The Burkina Faso international is an attacking midfielder, and joins the Stamford Bridge club from Association Jeunes Espoirs De Bobo-Dioulasso in his home country.
The 18-year-old has penned a four-and-a-half year contract with Chelsea in October, but has now signed the deal with the transfer window now reopened.
Traore, who made his international debut days before his 16th birthday in 2011, impressed during a trial period with the club on their pre-season tour of Asia in the summer.
And the teenager credits Mourinho as the major influence in his decision to sign for the club:
Jose Mourinho spoke a lot during training and he is one of the best managers, maybe the best in the world.
He is a very good person and it is a dream for me to work with him.
I can now say I am living a dream.
Traore is now relishing the opportunity to play alongside some of his heroes:
They're very good players and I am trying to be like them.
At Chelsea I look to follow the way of some African legends like Didier Drogba, Michael Essien and now Samuel Eto'o.
The news of Traore's arrival comes at the same time as Chelsea announced an annual loss of £49.4 million for the financial year, which could signal the end of free-spending at the club, according to Jeremy Wilson of the Daily Telegraph.