The Basel winger has signed a five-and-a-half-year contract at Stamford Bridge and will wear the No.15 shirt for the Blues, who fought off competition from Liverpool
The Stamford Bridge club looked to reinforce their attacking options after it became clear Juan Mata was set on a move to Manchester United, and have moved for the Egypt international after his impressive performances for the Swiss outfit.
The 21-year-old, who has signed a five-and-a-half year deal, was the subject of heavy interest from Liverpool before the Blues forced through a deal in midweek.
Salah told Chelsea's official website: "I'm very happy to sign for such a big club. I hope I can make the Chelsea supporters happy."
Salah has shone for Basel in the Champions League this season, and scored a late chipped winner against the Blues in November to hand his side a 1-0 victory over Mourinho's men.
With Mata completing his £37.1 million move to Old Trafford on Saturday evening after failing to fit into Mourinho's first-team plans this season, replacement Salah will offer the Portuguese coach width and penetration from the flanks.
Liverpool boss Rodgers bemoaned losing out to Chelsea for the winger's signature, telling talkSPORT: "It is one where the agent of the player and the clubs were talking but if the deal can't be done, the deal can't be done."