The hosts started aggressively, with Ibrahim Sulemana's shot from outside the box after three minutes going just wide of the target.
Italy striker Kean struck on the hour mark when he ran into the box and Montipo smothered his shot but could not stop it from tickling into the net.
The visitors had a scare five minutes from time when Verona were awarded a penalty after a high boot from Leonardo Bonucci, but the decision was changed after a VAR check.
Juventus extended their winning streak in the league to five games while Verona have now suffered nine Serie A losses in a row in for the first time.
Juventus are third on the table with 28 points from 14 games, with Lazio to play later on Thursday. Verona are bottom of the table with five points.