Salah had been a rumoured January transfer target for Brendan Rodgers' Liverpool, although they were usurped by Chelsea's bid rumoured to be in the region of £18 million.
The 21-year-old has come to prominence with a series of impressive displays against the Premier League club in recent seasons, having scored against them in European competition last season and this.
He commented: "I'm very happy to sign for such a big club. I hope I can make the Chelsea supporters happy."
Salah's arrival at Stamford Bridge goes some way to off-setting the departure of Juan Mata - who completed a £37.1m move to Manchester United on Saturday.
Although he has not played since mid-December, Salah - who has scored 17 goals in 27 games for Egypt - may be in line to make his Chelsea bow when Jose Mourinho's side welcome West Ham to Stamford Bridge on Wednesday.
He is the club's second January signing after Nemanja Matic returned to Chelsea from Benfica, while Kevin De Bruyne has joined Mata in leaving the London club with Michael Essien also set to complete a move to Milan.
Mourinho's men are third in the Premier League, two points off leaders Arsenal, and made it into the FA Cup fifth round on Sunday courtesy of a 1-0 home win over Stoke City.
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