Having been sacked by Milan in January, Allegri was the surprise choice to replace Antonio Conte at Juve but victory at San Siro sees him extend his 100 per cent record in charge of the defending champions.
Juventus dominated possession throughout the match with Roberto Pereyra, back after European suspension, particularly impressive.
They struggled to find the finishing touch, though, and had to wait until the 71st minute for Tevez to strike the decisive blow.
Paul Pogba held off three Milan defenders on the edge of the area before playing an exquisite pass to the Argentine and he fired beyond Christian Abbiati.
Milan introduced loanee Fernando Torres in the final 15 minutes but, like the rest of his team-mates, he was unable to find his way past Juve's solid defence.
Elsewhere on Saturday, Empoli came from two goals down to earn a 2-2 draw at Cesena.
The hosts settled the quicker at the Stadio Dino Manuzzi and opened the scoring on the half hour as Guido Marilungo found the back of the net at the second attempt.
Franco Brienza saw his effort pushed away by Luigi Sepe and although Marilungo struck the bar, he was able to head home the rebound.
Gregoire Defrel doubled their lead two minutes later as he unleashed a 25-yard strike that bounced in front of Sepe and flew into the corner.
Cesena's first-half dominance did not continue after the break, though, as Empoli were awarded a penalty on 55 minutes that Francesco Tavano duly converted.
The comeback was complete when Daniele Rugani struck with 18 minutes to play and earn Empoli a first point of the season.
