Stand-out recruits Gonzalo Higuain and Steven N'Zonzi each had goals ruled out via VAR reviews for their respective sides after visiting defender Federico Fazio cancelled out Franck Kessie's 40th-minute opener.
It was a frustrating night in front of goal for Higuain but the ex-Juventus man turned provider in the 95th minute, threading a fine pass for Cutrone to slot home.
Gennaro Gattuso led raucous celebrations as his team bounced back from last weekend's chastening 3-2 loss at Napoli, while Eusebio Di Francesco's Roma are now winless since the opening weekend.
Milan right-back Davide Calabria fizzed just over Robin Olsen's crossbar before Roma's Sweden international goalkeeper was called upon to thwart Higuain and Suso.
The breakthrough arrived when Ricardo Rodriguez proved too nimble for Fazio and Kessie caught the left-hand side of Roma's defence napping – stealing in for a close-range finish.
Higuain sloppily lost possession for his opposite number in Roma's attack, Edin Dzeko, to curl wide in the 49th minute.
The pressure was building on Milan and Fazio atoned for his part in the opener with a fine volleyed finish when the ball was worked back into the box after Gianluigi Donnarumma punched poorly from a corner.
The Rossoneri thought they had an instant response but replays showed Higuain was fractionally offside when he scampered in between Fazio and Kostas Manolas to finish coolly.
N'Zonzi was the next man to fall foul of technology, although the shaky set-piece defending on display from the hosts made Roma appear the most likely winners. Cutrone had other ideas.
