"Gattuso is a legend, is a big honour to have him as coach," said Cutrone.
"Since I was a child I have appreciated him for his grit, the way he helped his team-mates."
On his progress under his former boss, Cutrone added: "I honestly did not expect to play so much. I will always be grateful to Montella.
"The experience in these games helped me a lot."
Gattuso will take charge of the team for the first time at bottom club Benevento on Sunday and Cutrone, who replaced Nikola Kalinic as a second-half substitute in the 0-0 draw with Torino before Montella's dismissal, expressed his desire to get back to winning ways.
"I don't like to lose," he said. "I want to win even during the training matches."
I don't think it's possible to not like this guy