The 18-year-old agreed to join the Blues in October 2013 from Association Jeunes Espoirs De Bobo-Dioulasso in his home country and has now put pen to paper on a four-and-a-half year deal.
Traore said: "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.
"My favourite players are Juan Mata, Mesut Özil and Eden Hazard. 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."