Following the opening round's 3-2 defeat at Napoli when they threw away a 2-0 lead, Milan appeared set to suffer another disappointing night before Cutrone converted a defence-splitting pass by Gonzalo Higuain.
Both sides had a goal disallowed after referee Marco Guida consulted the Video Assistant Referee (VAR), Higuain seeing his effort scrapped for offside while Roma's Steven Nzonzi was adjudged to have handled the ball.
With rumours aplenty that former Chelsea manager Antonio Conte was being lined up to replace Gattuso in case of a prolonged poor start to the season, the former Milan enforcer was able to breathe a huge sigh of relief after the final whistle.
Determined to make amends for the slip-up at Napoli, Milan laid siege to Roma's goal in the opening half as Roma keeper Robin Olsen denied Higuain and Suso before Franck Kessie broke the deadlock in the 40th minute.
The Ivorian midfielder blasted the ball home from close range at the far post after good work by left back Ricardo Rodriguez, who turned his marker and delivered a clever pass across the face of goal.
Subdued and struggling to carve out anything up front, Roma equalised out of the blue just before the hour as centre back Federico Fazio unleashed a stinging volley from inside the penalty area after Milan failed to clear a corner.
Higuain, who is on loan at Milan from champions Juventus, thought he had scored his first goal for the team only to see it cancelled after a lengthy VAR analysis, with Nzonzi suffering the same disappointment at the other end.
With time running out as Olson made two more good saves in the second half, Cutrone delighted the home fans when he swept the ball past the Sweden keeper after replacing Giacomo Bonaventura in the 84th minute.
In Saturday's standout clash, Juventus visit promoted Parma with their stellar new signing Cristiano Ronaldo still looking for his first goal for the Italian title holders.
Inter Milan are away to Bologna in Saturday's other fixture while Sampdoria are at home to Napoli and surprise package SPAL visit Torino on Sunday.
(Writing by Zoran Milosavljevic; Editing by Clare Fallon)
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