Central defender Medhi Benatia struck twice for Juventus and Douglas Costa was also on target on an unhappy evening for Milan goalkeeper Gianluigi Donnarumma who was at fault for the second and third goals.
Juventus has won a total of 13 Italian Cups, at least four more than any other team.
Milan coach Gennaro Gattuso said that he is treating the match against Bianconeri as a World Cup final.
Massimiliano Allegri's side are one point away from claiming a seventh consecutive Serie A title with two games to play and could seal it this weekend against Roma in the same Stadio Olimpico stadium.
In the final, which was held in Rome, Juventus defeated Milan 4:0.
After a slow-paced first half, Juventus took control when Benatia headed in from a corner.
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The tempo of the match was fizzling out in the closing minutes of the first half as the sides went into the break goalless. Donnarumma reacted too late to stop Brazilian winger Costa's shot from the edge of the box in the 61st. In the second leg, Pjanic scored a penalty 15 minutes from time when Blaise Matuidi was fouled in the area to hand Juve a 2-0 win on aggregate.
Kalinic, who has not reached expectations at Milan, then added to the misfortune by finding his own net with his back to the goal by the near post as he attempted to defend a corner.
While probably not the most memorable of games between the two sides, the result was significant in that it meant that Juventus became the first team in Italian football to complete back-to-back doubles.
"We've written another chapter in Juventus' history and now is the time to celebrate", Allegri was quoted as saying by Juventus' website. They showed technical quality and mental strength in the game, gifting another night to keep in mind to the fans.
"Today I feel this incredible pressure on me, as if the fans were on my shoulders", said the 40-year-old.
"It's not that Juve have to win, but I'm now unfortunately used to the idea that people take our victories for granted".