The Croatian boss replaced Davide Ballardini before the global break for his third spell in charge of Genoa and he made an immediate impact as the Italian champions failed to record a victory for the first time in nine league games this season.
But Genoa enjoyed the better of the second half and got their reward after 67 minutes when Bessa headed home.
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With Genoa struggling to contain the lively Cancelo down the right Juve were firmly in control, their only regret at the break being they weren't further in front.
Juve are now four points clear on top of the table ahead of second-placed Napoli.
But Daniel Bessa equalised with a header from Christian Kouame's cross.
Allegri introduced Paulo Dybala off the bench and both he and Miralem Pjanic fired narrowly wide from distance, but Genoa withstood the late pressure to move 10th in the table and give Napoli the chance to close the gap to the leaders.
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With the pressure off, Marko Rog then added a third with five minutes to go.
Earlier, Roma came unstuck 2-0 to visiting minnows SPAL to dash their hopes of a fourth straight league win. Every match has to be won on the pitch in 90 minutes.
Ronaldo is primed to start Juventus' Champions League showdown with Manchester United, having missed their last European encounter against Young Boys through suspension.
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