The Bianconeri wrapped up the Scudetto earlier this month, and Massimiliano Allegri added another piece of silverware to the cabinet to complete back-to-back doubles since taking charge in Turin.
Alvaro Morata scored the winning goal for Juve, with the Spaniard coming off the bench and breaking the deadlock in the second half of extra-time to leave Milan supporters heartbroken.
The Rossoneri made the more impressive start to the game of the two sides, and enjoyed plenty of possession throughout the first half.
Having survived an early Paul Pogba penalty shout, Giacomo Bonaventura twice went close to breaking the deadlock, while Neto almost misjudged an effort from the Italian which would have resulted in an embarrassing error.
Mattia De Sciglio was next to try his luck from distance, while Andrea Poli blazed over soon after when he should really have done better, with Juve struggling to keep hold of possession and in turn pose a threat in the final third.
Cristian Brocchi would have been pleased with what he saw from his side in the opening 45 minutes, but naturally there would been concern within the Milan camp that they weren’t able to take advantage and make the most of their chances.
The second period began much like the first ended with Juraj Kucka and De Sciglio striking from range as Milan continued to push forward.
While the Bianconeri responded momentarily, the game soon settled into a familiar pattern with Milan’s desire and energy seeing them enjoy the lion’s share of possession and threatening to find the opening goal of the encounter.
Allegri’s men began to take control in the latter stages, but much like Milan, they were unable to find a decisive touch in the final third to make the difference with a frustrated Pogba leading their charge.
However, neither side were able to find a breakthrough before the end of normal time and that continued through the opening period of extra-time as Bacca and Pogba both went close again but couldn’t find the back of the net.
The deadlock was finally broken with 10 minutes remaining in extra-time though, as substitute Morata kept his composure to produce a classy finish from a Juan Cuadrado cross to seal the victory.