The Italy midfielder scored his first with a lucky ricochet with three minutes to go after playing a one-two with Mirko Vucinic before making the game safe in injury-time, firing a 25-metre volley through goalkeeper Christian Abbiati's legs.
"We deserved to score a long time before," Juve coach Antonio Conte told Italy's Sky Sports TV.
"Our intensity and approach were brilliant. We dominated possession, which was very important against a side like Milan, but the best thing that came from tonight was the way we played as a team."
The once mighty Turin side, seventh the last two terms, are unbeaten under new boss Conte with three wins and two draws.
"I'm not looking too far ahead," added the ex-Juve player. "The only promise I will give is to make the team competitive because Juve deserve to be challenging among the best."
Earlier on Sunday, Udinese beat Bologna 2-0, while 2-0 victories for Cagliari and Palermo over Lecce and Siena helped the island teams join Napoli within a point of the leaders.
A Sebastian Giovinco brace helped Parma beat Genoa 3-1 and Lazio grabbed the three points at Fiorentina thanks to a late Miloslav Klose header in a 2-1 win.
Encouraged by a noisy crowd, Juventus edged a fast-paced opening quarter with ex-Milan midfielder Andrea Pirlo coming closest with two trademark long-range swervers that Abbiati had to watch closely.
Juve's Vucinic, playing as the lone striker in Conte's 4-5-1 formation, came close to giving the home side the lead with two sublime pieces of skill just after the half-hour mark.
First, on 33 minutes, the Montenegrin had Abbiati scrambling to his left to save after turning sharply in the box and fizzing a right-foot shot towards the corner.
Then, three minutes later, the ex-AS Roma man brought the Turin crowd to their feet as he killed the ball 20 metres out and bent a dipping shot against the underside of the Milan bar.
With the champions struggling to contain the inspired forward, Vucinic's next trick was to shift the ball onto his favoured right foot and fire a raking drive just wide of the upright as Juve pressed for the all-important first goal.
Gianluigi Buffon came to Juve's rescue six minutes after the restart, stretching out his giant hand to turn aside Boateng's angled volley as a more aggressive Milan attempted to up the pace.
Vucinic threatened again on 67 minutes after latching onto a misplaced Mark van Bommel back-pass but the lightning fast Thiago Silva recovered to save the Dutchman's blushes.
SECONDS REMAINING
Juventus continued to dominate and Chilean midfielder Arturo Vidal saw two long-range efforts flash wide as the pace slowed with a quarter of an hour to go.
On 80 minutes Vucinic thought he had won it but could only hold his head in his hands as his shot from ju
