Mandzukic, 29, only joined Atletico from Bundesliga side Bayern Munich in July 2014 and went on to score 20 goals in all competitions for the Spanish club last season.
The former Wolfsburg man shunned interest from the Premier League club to put pen to paper on a four-year deal with the beaten Champions League finalists.
As expected, the Croatia international passed Monday's medical leaving Juventus to conclude the €19m deal - split in instalment payments over three years - with €2m in bonuses to be potentially paid to Atletico.
Mandzukic's arrival comes amid speculation that Carlos Tevez could leave Juventus, with boyhood club Boca Juniors expected to sign the Argentine after the ongoing Copa America concludes.
The transfer of Mandzukic marks the second move Juve have made to bolster their forward this month line,following the signing of Pablo Dybala from Palermo for a fee that could rise to €40m.