Italian media report that the extension until June 2023 was automatic after Cuadrado played his 40th match of the season at Cagliari last weekend.
Juve are also reportedly in negotiations for a longer deal for Cuadrado, who was out of contract at the end of this campaign, which would keep him at the club until 2024.
That deal would also spread his five million euro ($5.4 million) salary across two seasons in an attempt to lower costs.
Juve recently decided to release playmaker Paulo Dybala for free in the summer as part of the same strategy.
Colombia international Cuadrado moved to Juve from Chelsea in 2015 and has won five Serie A titles and four Italian Cups since, playing 262 times for the Turin giants in all competitions.
The 33-year-old has played consistently as both a full-back and winger for Juve and is a key player for coach Massimiliano Allegri, who returned to the Allianz Stadium last summer.
Juve are currently six points behind league leaders AC Milan with six games remaining and should qualify for the Champions League next season.