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.
An own goal by substitute Nikola Kalinic completed the drubbing.
Asamoah, who is in his sixth year at Juventus, took part in the final, lasting the full duration.
Juventus boss Allegri elected to keep regular Brazilian left-back Alex Sandro on the bench and put his faith in Asamoah and other than one big mistake, the 29-year-old put in a good shift.
He will become Serie A champion for the sixth consecutive time if Juventus earn a point against AS Roma the weekend.
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