While the 19-year-old Donnarumma could not be blamed for Medhi Benatia's opener in the final, his mistakes led to the ensuing goals from Douglas Costa and Benatia.
Their cause was not helped by two glaring errors from highly rated goalkeeper Donnarumma, as Milan conceded four in the space of 20 astonishing minutes.
Blaise Matuidi was fortunate not to score an own goal, slicing a cross against the woodwork, but luck deserted Kalinic at the other end, his header glancing past Donnarumma to round out a thoroughly miserable evening for Milan.
Juventus became the first club to win the Italian Cup four times in a row.
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.
After a slow-paced first half, Juventus took control when Benatia headed in from a corner. Two minutes later, Benatia got his second of the night to make it 3-0 for Juventus.
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Both sides have different reasons to claim the piece of silverware as Serie A champions-elect Juve aim to complete an unprecedented fourth domestic double on the trot, while this is Milan's first real shot at a trophy under the Chinese management.
Juventus added a fourth thanks to an own goal from Nikola Kalinic on a thoroughly miserable night for Donnarumma and Milan. We'll do this with Juventus administration.
"The guys had an extraordinary game and deserved this win, taking nothing away from Milan, who had a good first half", Allegri said.
"Winning our fourth Coppa Italia in a row wasn't easy after a hard and sapping season".
"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", said Allegri, 50, after achieving his tenth coach as a coach between Juventus and former club AC Milan.