The team won 4-2 on penalties against Juventus after a 0-0 draw to win their first trophy in six years and Gattuso believes his side earned their victory.
The 42-year-old manager told Rai Sport: "Whoever does this job must have respect for it.
"We are lucky. This is why I get angry; there must be passion, respect, seriousness. Then there is a God of football who rewards you when you work well."
Gattuso has now urged the club to use the cup win as a springboard for the remainder of the Serie A season.
They are currently in sixth place and are nine points from a Champions League spot.
He said: "We have a duty to play these last 12 games with respect. We must not think about giving up."
Paulo Dybala and Danilo missed their penalties for Juve in the defeat, meaning Maurizio Sarri has now lost two finals since taking charge of the Turin club.
The 61-year-old coach added: "There is disappointment for the boys, for the club, for the fans, but at the moment we cannot express more."
Sarri feels the team are yet to perform at the best since football resumed in Italy following the coronavirus pandemic.
He said: "We have tried to press less high, our condition is not the optimal one, but we did the game with excellent levels of application.
"I think that at the moment we are struggling especially in terms of individuality and that this lack of brilliance is characteristic of this period."