
Kean headed in the first goal in the eighth minute after a well-timed cross from Weston Mckennie. Mattia Valoti equalised for Monza in the 24th minute with another header, this time from a corner.
Substitute Chiesa sent Juventus ahead again when he managed to move into the box and placed the shot into the far right post.
Monza desperately tried to get an equaliser but the Juve defence managed to control the pressure until the end of the match.
Earlier on Thursday, Atalanta thumped Spezia 5-2 while Lazio beat Bologna 1-0 reach the quarterfinals.
Coppa Italia quarterfinals:
Inter v Atalanta on Jan 31
Fiorentia v Torino on Feb 1
Roma v Cremonese on Feb 1
Juventus v Lazio on Feb 2