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A relatively quiet game offered little in the way of clear-cut opportunities, Danilo hitting the frame of the goal being the best chance of the first half.
However, Kean ensured the home fans would have something to celebrate after capping off a fine Juve move.
Far from a vintage performance, Juve has moved four points behind fifth-placed Atalanta and three behind sixth-placed Roma, which plays on Monday (AEST).
Verona almost broke the deadlock after 15 minutes, Kevin Lasagna pulling a dangerous cross to the far post for Fabio Depaoli to smash a volley narrowly wide of Wojciech Szczesny's goal.
Danilo came closest before the break, his free-kick clipping the top-left corner of the crossbar.
Juve got its winner in the 55th minute. An excellent move culminated in Manuel Locatelli playing in Kean, who smashed home to take his tally of league goals for the season to six.
Massimiliano Allegri's side was not home and dry, however, with Szczesny charging off his line to push a loose ball away with Lasagna lurking after his defence left him in the lurch.
Juve had Szczesny to again thank when he kept out Filippo Terracciano's fierce effort.
Bremer squandered a golden opportunity to wrap up the points late on, somehow bundling an attempt over from inside the six-yard box, though fortunately for the defender, Verona could not make Juve pay for his miss.
