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."
"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."
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.
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.
