Victory not 'revenge' for Allegri
The 47-year-old returned to Milan for the first time to face his former club and masterminded an impressive Juve display that was capped off by Carlos Tevez's winning goal.
Numerous chances fell the way of the visitors before Tevez's strike - his second Serie A goal of the season - while Milan were only able to fully test Gianluigi Buffon with a first-half header from Keisuke Honda.
Allegri was delighted with the victory, which extends their winning start to the campaign, but insisted the three points was most important and not inflicting defeat on his previous employers.
"After three and a half years at Milan sharing emotions, intense moments and victories, it would've been impossible to consider this a normal game," he told Sky Sport Italia. "Having said that, it was absolutely not a chance for revenge.
"Milan was an important step in my career and I am still very fond of this club. Now I am at Juventus and I am focused on winning with Juventus.
"I think it was a good game.
"Milan defended, we created several scoring opportunities and ran some risks on the counter-attack, but overall we controlled the match pretty well.
"We are on a journey and will improve as time goes on. There are still things to fix and do better on, but the players mainly controlled the ball well after taking the lead and I liked their attitude."
As well as maintaining their winning run, Juventus' clean sheet means they remain yet to concede a goal in all competitions.