The Blues announced on Thursday night the signing of the 21-year-old Egyptian winger, subject to the agreement of personal terms and the signing of a medical.
Chelsea confirmed the deal has been concluded before Sunday's FA Cup fourth-round tie with Stoke.
The deal is for a reported fee of £11million as Chelsea signed the forward ahead of Liverpool.
Once it was clear Juan Mata was to join Manchester United, Chelsea boss Jose Mourinho began looking for a replacement and Salah immediately came to mind.
Salah has played against Chelsea four times in the last 12 months, scoring three times, including in Basle's two Champions League Group E victories over Mourinho's men earlier this season.
Speaking on Friday, Mourinho spoke of his admiration for Salah, the potential of the player and said the Blues made their move quickly.
"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.
"We believe if he comes that he will have a similar process to the Willian one. Willian needed two or three months to grow up and to feel comfortable playing for us (following his August signing from Russian club Anzhi Makhachkala).
"We think with him and (Andre) Schurrle, (Eden) Hazard, Oscar and Willian we will be fine.
"We lose an experienced top player like Juan, but we bring in a young player with great potential.
"Willian did that transition from his previous world to the top world of football. Hopefully, if Salah comes, he can do the same.
"We are ready to help, we are ready to support.
"He has some experience in Europe, the Europa League, Champions League, so he's not a naive boy arriving in the jungle.
"Hopefully he comes and hopefully with the talented players we have around in the same positions we all can develop together."
Mourinho believes Salah is joining the right club to grow as a person and as a player after he was caught up in controversy during his time at Basle.
During a two-legged Champions League qualifier between the Swiss side and Maccabi Tel Aviv last August, Salah refused to shake hands - instead changing his boots in one instance and bumping fists with his opponents in the other - while there were also comments attributed to him expressing an anti-Israeli sentiment.
Salah insisted the comments were fictitious and his actions were a result of external pressures.
Yet Mourinho has promised to nurture the 21-year-old at Chelsea, who are owned by Russian billionaire Roman Abramovich, who is Jewish.
"It's not a concern, because we know the club Chelsea is, I know the manager I am," Mourinho said.
"I know that Chelsea's capable of helping people and making people feel the way we think people should feel, which is in football and in life be open and respectful to every ethnicity, to every religion.
"People has the right to be what they are. Chelsea as a club, and me as a manager, we will be more than ready to help him."
Salah has signed a five-and-a-half-year contract and will wear the number 15 shirt for Chelsea.
He said: "I'm very happy to sign for such a big club. I hope I can make the Chelsea supporters happy."