Alves arrived in Turin on Sunday and underwent his medical with the club on Monday morning.
Barcelona recently already announced that the Brazil international would be allowed to leave on a free despite a contract until June 2017, with Paris Saint-Germain and Juve keen to sign the right-back.
Alves has opted to snub a move to PSG, though, and will continue his career with the Serie A champions having signed a two-year deal, with the option of a further 12 months thereafter.
The 33-year-old started his professional career with Bahia and also wore the jersey of Sevilla before joining Barcelona in July 2008.
He won 23 trophies during his time at Camp Nou, including six La Liga titles, three Champions League trophies and as many Club World Cups.