The lead was doubled on 61 minutes as Douglas Costa's strike was spilled into the net Donnarumma, who should have done better with the effort. Then Kalinic headed a corner into his own net.
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.
After the final whistle, Milan captain Leonardo Bonucci and Juventus counterpart Gianluigi Buffon attempted to console Donnarumma.
It was the first of two titles that Juventus is hoping to celebrate over the space of four days at the Stadio Olimpico.
Massimiliano Allegri said he is not thinking about his future as the in-demand Juventus head coach prepares for end-of-season talks amid Premier League links.
Juventus, who became the first team to win the competition without conceding a goal, are nearly certain to add this year's Serie A title to their trophy cabinet and complete a fourth successive double.
And former AC Milan midfielder Gattuso conceded he felt huge pressure as he bids for his first coaching title after taking over last November.
Morocco global Mehdi Benatia broke the deadlock for the hosts 10 minutes into the second stanza with a free header inside the box, 1-0.
Milan striker Nikola Kalini completed the rout with an own-goal.