The 27-year-old, who made little impact in 15 Chelsea appearances including 11 as a substitute, has previous experience of Serie A, having played for Fiorentina, Lecce and Udinese.
Juventus said on their website (www.juventus.com) that Cuadrado, who was given a warm welcome from fans when he flew to Turin on Monday for a medical, would join his new team mates for training on Tuesday.
"Juventus are a great team and one of the best in Europe," he told the club website.
"We've got everything it takes to fight for the Scudetto and we hope to defend our title. I'm here to give 100 percent, work hard and prepare well for every game."
An important player for Colombia at last year's World Cup in Brazil, Cuadrado joined Chelsea from Fiorentina in February on a four and half year deal for an estimated 23.3 million pounds.
His move follows the arrival at Stamford Bridge last Thursday of Spain winger Pedro from Barcelona.
(Reporting by Alan Baldwin, editing by Ed Osmond)
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