Rafael Leao was the star of the night, scoring the opener with a thunderous drive before dedicating his goal celebration to the memory of former Portugal teammate and Liverpool striker Diogo Jota.
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Leao also laid on Ruben Loftus-Cheek's second-half goal, with Noah Okafor scoring twice late on to round out Milan's scoring.
For Liverpool, Dominik Szoboszlai and Cody Gakpo struck, though Milan were the superior team on the night.
Allegri handed new left-back signing Pervis Estupinan a debut, while Florian Wirtz featured for Liverpool. Reds manager Arne Slot also started the game with Rio Ngumoha on the flank.