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Verona had lost eight consecutive league games while Massimiliano Allegri's men had won four in a row.
However, Salvatore Bocchetti's men made it difficult for Juve and it looked as if they had a chance of earning an important point.
But Kean's deflected 60th-minute strike made it three goals in three starts for him and his team held on to continue their winning streak, despite a late red card for Alex Sandro.
Verona started brightly and went close within three minutes, though Sulemana's powerful drive flew just past the post after a corner was cleared to him outside the box.
Juve grew into the game, with Arkadiusz Milik and Manuel Locatelli both testing Verona goalkeeper Lorenzo Montipo.
Pawel Dawidowicz missed a decent opportunity after the interval when he scooped over after the ball fell kindly to him in the area.
Juve nearly went ahead when Adrien Rabiot crossed for Kean, but the low delivery just evaded the former Everton striker's outstretched boot.
The game was becoming more stretched, and Juve hit the hosts on a sublime counter to grab the lead on the hour.
Kean took Rabiot's pass in his stride before his deflected effort was pawed at by Montipo, but not enough to stop it bouncing in.
Juve had an almighty scare when a collision between Leonardo Bonucci and Simone Verdi in the box led to the referee pointing to the spot, but a VAR check encouraged him to reverse his decision.
Sandro was shown a second yellow card in injury-time for cynically barging over Kevin Lasagna as the forward charged down on goal, but it did not matter as the resulting free-kick flew over and the final whistle eventually came, much to Juve's relief.