Media reports said Chelsea had paid a fee of around 35 million euros (26.40 million pounds) for Cuadrado.
Egyptian Salah, who has struggled to establish himself at Chelsea since joining from Swiss club Basel last January, has agreed to join Fiorentina on loan.
Cuadrado had been on Chelsea's radar for some time and the deal was finalised after the Stamford Bridge outfit sold Germany winger Andre Schuerrle to Bundesliga side VfL Wolfsburg earlier as the transfer deadline loomed.
"I am very happy and thankful for this opportunity I've been given," the 26-year-old told the club's website (www.chelseafc.com)
"This is a great club and honestly it is like a dream to join the Chelsea family and to know that the manager believes in me.
"I'm happy. It's really going to be great to work with one of the best coaches in the world."
Cuadrado has been a key figure for Fiorentina since joining from Italian rivals Udinese in 2012 and his departure will be a blow to their hopes of qualifying for next season's Champions League as he leaves them in sixth spot.
The signing of 22-year-old Salah will soften the blow, especially if he can rediscover the form that prompted Chelsea to splash out 11 million pounds for his services.
Comfortable on either flank, or behind a striker, Cuadrado wil offer Chelsea manager Jose Mourinho more attacking options as the season reaches a critical phase.
Chelsea are five points clear in the Premier League, in the League Cup final against Tottenham Hotspur next month and have a Champions League last 16 tie against Paris St Germain.
Cuadrado scored 26 goals in 106 appearances for Fiorentina and has 37 caps for Colombia, having helped them to the quarter-finals of the 2014 World Cup.
(Editing by Tony Jimenez)
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