The Rossoneri saw Franck Kessie's opener cancelled out by Federico Fazio, while both sides had goals disallowed using VAR in the second half.
With 30 seconds left on the clock, Cutrone ran on to Gonzalo Higuain's pass to beat Robin Olsen.
“I am so happy. We won the game, what more could you want? A victory like that is even more enjoyable," Cutrone told Sky Sport Italia.
“I couldn't hear or understand anything once the ball went in. I hope to continue like this, help the team and keep us going strong.
“We showed good football this evening again, got the deserved three points and above all earned it with good football.
“Higuain is a great striker, as the Coach said one of the best in the world. It's an honour to play with him and to learn from him.
“I never like being on the bench, obviously. I know there's a champion in front of me, but I'll do everything I can to make life difficult for the Coach and help the team any way I can."