An early Cristiano Biraghi own goal was the difference as Juventus' title defence got off to a bright start at the Stadio Marc'Antonio Bentegodi.
Defender Biraghi turned Martin Caceres' header past his own goalkeeper with just seven minutes on the clock, and Juve should have helped themselves to more.
Arturo Vidal struck a post and saw an effort cleared off the line, while Carlos Tevez twice hit the crossbar.
Chievo could have stolen an undeserved point late on but Maxi Lopez was denied by Gianluigi Buffon.
Allegri is under pressure to continue the success achieved by predecessor Antonio Conte, who stood down after winning three consecutive league crowns and has since been appointed Italy coach.
Roma, runners-up to Juve last term, also got their campaign off to a winning start as Radja Nainggolan's first-half strike and Gervinho's goal after the break earned them a 2-0 success.
Rudi Garcia's side kept 21 clean sheets in 2013-14 and continued that trend at the Stadio Olimpico against a Fiorentina outfit who finished fourth last time out.
Nainggolan, who scored the winner at Fiorentina in April as Roma completed a double over Vincenzo Montella's men, pounced after Gervinho's effort had been parried into his path by Neto.
Josip Ilicic struck the crossbar on the hour mark, but that was as close as the visitors went to scoring.
And Gervinvho grabbed a deserved goal with the last kick of the game after confidently rounding Neto.
