The reigning champions began their title defence by continuing their winning streak in the league from last season with an eighth-straight win, but the margin of victory should have been greater.
Juve were well into their stride early on and went ahead after seven minutes when Martin Caceres headed Carlos Tevez's deep corner in via the unfortunate Cristiano Biraghi.
They dominated the first half, as Arturo Vidal saw an effort cleared off the line before hitting the post with a diving header, and Tevez rattled the crossbar twice in as many minutes towards the end of the half.
But, for all their dominance, Juve were nearly made to pay towards the end when the ball fell invitingly to Maxi Lopez 12 yards out and Gianluigi Buffon had to make a terrific sprawling save.
Although Allegri was satisfied with Juve's performance in his first game as coach, he warned that the goal return had to be better.
He told Sky Sport Italia: "More than angry I was a bit worried, because after wasting so many chances we risked an equaliser with the only Chievo chance and Buffon made a great save.
"The team played well, though we need to improve our finishing. At this moment the team is working hard, it was important to win the opener ahead of the break for international duty."
Juve handed a debut to French midfielder Kingsley Coman after his close-season move from Paris Saint-Germain and he delivered an impressive performance that enthused Allegri.
"The kid Coman played like a veteran," he added. "He has great quality and character. He’s a young lad and we should let him grow, but he has real talent."
