Making his first return to San Siro since being sacked in January, Allegri watched his new side create a handful of chances, but he had to wait until the 71st minute for the decisive goal.
The visitors were impressive in the opening half as Roberto Pereyra and Giorgio Chiellini both had sights of goal, while Claudio Marchisio saw a low shot come back off the post.
Milan struggled to contain the defending champions and eventually succumbed to Tevez's goal, following a superb Paul Pogba pass.
Fernando Torres' introduction for his debut with 15 minutes left to play was not enough to spark Milan in to life and they slipped to their first competitive defeat under new head coach Filippo Inzaghi.
As in their UEFA Champions League win over Malmo on Tuesday, Juventus dominated possession but were left frustrated for long spells.
Milan's back four remained resolute throughout the first half, with only Pereyra - who returned after his European suspension - going close early on with a sixth-minute effort that curled just over.
For all Juve's control, it was the hosts who were first to test the opposition goalkeeper, as Keisuke Honda brought a smart save out of Gianluigi Buffon after 26 minutes.
Having drifted away from Chiellini, the Japan international was superbly found by Sulley Muntari’s cross-field pass, but his close-range header was parried away.
That appeared to breathe new life into Allegri's side and Christian Abbiati - who came in for the injured Diego Lopez - was twice called into action to deny Fernando Llorente and then Pereyra in the space of four minutes.
Juventus were slowly building up momentum and Chiellini wasted their best opening of the half when, having timed his run perfectly to meet Marchisio's chipped pass, he failed to connect with a header from six yards.
Marchisio was beginning to have a significant impact and the midfielder was unlucky not to open the scoring before the break when his low effort struck the post with Abbiati well beaten.
The second half followed the same pattern as the first, with the ball stuck in midfield in the opening 15 minutes and neither side able to find an opening.
That changed on the hour, though, as a Pereyra delivered a low corner to the near post, where Marchisio - who had evaded his marker - could only send a low shot wide of goal.
Milan's impact on the game continued to dwindle and Juve began to turn the screw, launching wave after wave of attack.
And with 19 minutes to play their pressure finally paid off, as Pogba delivered an exquisite pass into Tevez, who showed great feet to net his second league goal of the season.
In an attempt to change Milan's fortunes, Inzaghi introduced Giacomo Bonaventura, Giampaolo Pazzini and Torres - the latter for the first time since his loan switch from Chelsea - but the trio struggled to make an impact as Juventus dictated the closing stages.
Jeremy Menez thought he should have had a penalty late on following a challenge from Chiellini, but his and Milan's appeals were waved away as Allegri celebrated.
