The Turin side led 1-0 for almost the entire game only to see Valentín Carboni equalize for Monza a minute into injury time.
However, the visitors reacted with aplomb and three minutes later Federico Gatti converted a cut-back from France midfielder Adrien Rabiot to give Juventus the lead again — and all three points.
The main action of the game until then came after 12 minutes. Monza goalkeeepr Michele Di Gregorio saved his team twice in quick succession, first from Dusan Vlahovic's penalty and then again brilliantly from the rebound.
However, Rabiot rose to meet the resulting corner and power an unstoppable header into the net to give Juventus the lead.
The Turin giant is now unbeaten in nine league games.
The result lifts Juventus into first place in Serie A. It's a point ahead of Inter Milan, which plays fourth-place and defending champion Napoli on Sunday.
Monza, which was promoted from Serie B last season and has surprised fans with its string of solid results, remains in 10th place with 18 points from 14 games.