The former Newcastle striker, who is the joint third highest Premier League goal scorer this season, will be eligible for Chelsea's FA Cup tie at Southampton on Saturday the club said.
"It feels good to be here, I'm very happy and very proud," the 27-year-old Senegalese forward said.
"When the club who won the Champions League wants you, the decision is very easy. This club is massive and that's something that helped the decision a lot. It was not a hard one."
Chelsea are currently fourth in the Premier League after losing at home to bottom club Queens Park Rangers on Wednesday.
(Reporting by Stephen Eisenhammer; Editing by John Mehaffey)