Salah has signed a five-and-a-half-year contract with the reigning UEFA Europa League champions and will wear the No 15 shirt at Stamford Bridge.
"I'm very happy to sign for Chelsea, such a big club in the world," Salah told Chelsea's official website. "I hope I can make the Chelsea supporters happy and have a good career at the club for many years to come."
Chelsea said on Thursday that they had reached an agreement to sign Salah, subject to the completion of a medical examination and the agreement of personal terms.
Manager Jose Mourinho revealed that the transfer had been completed during a television interview prior to his side's 1-0 win at home to Stoke City in the FA Cup fourth round on Sunday.
Salah has played against Chelsea four times over the last year, scoring three times, including in Basel's two UEFA Champions League group-phase victories against the London club earlier this season.
Speaking on Friday, Mourinho expressed admiration for the Egyptian international, who reportedly rejected overtures from Liverpool in order to move to west London.
"I like the fact that he can play the same way our attacking players do, which is players with adaptability, to play right, left, or behind the striker," Mourinho said.
"He's young, he's fast, he's creative, he's enthusiastic. When we analysed him, he looks the kind of humble personality on the pitch, ready to work for the team and to work and to adapt himself to a new life."
Confirmation of Salah's arrival comes less than 24 hours after Spain attacking midfielder Juan Mata completed his move from Chelsea to Manchester United for a fee of £37.1m.