Fofana, 20, joins Chelsea for a reported fee of £10.93 million ($13.15m) and becomes the latest January arrival at Stamford Bridge after Monaco defender Benoit Badiashile and Brazilian teenager Andrey Santos.
He has contributed 15 goals and five assists in 24 matches this season for Molde in the Eliteserien, the top flight of Norwegian football.
Fofana said in a statement: "Hello dear fans, I'm here, I've arrived well and I'm very happy to sign for the club of my dreams. I'll see you soon on the pitch, cheers!'
He has two international caps for Ivory Coast, with his debut coming in a 4-0 friendly win against Burundi in November.
Chelsea, who have struggled in front of goal this season, suffered a 1-0 defeat to Manchester City in the Premier League on Thursday, and face another fixture against Pep Guardiola's side in the FA Cup on Sunday.