The Basel winger was linked to both Chelsea and Liverpool during the January transfer window but has chosen a move to Stamford Bridge.
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 has 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.
"I'm very happy to sign for such a big club. I hope I can make the Chelsea supporters happy," Salah told Chelsea's official website.
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-pound 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."