"Dear Rino, I am happy to have spent almost two seasons with you. Thank you for your work and I wish you success wherever you go. A hug also to your wife and children," De Laurentiis wrote on Twitter after the game.
The announcement confirmed long-running media speculation that the former AC Milan manager would depart Naples when his contract expires in June.
It was followed by a tweet from the official club account that simply said: "Thank you, boss!" alongside a picture of Gattuso.
Gattuso succeeded Carlo Ancelotti as Napoli coach in December 2019 and led them to Coppa Italia glory in June 2020.
Although his departure was widely expected, Gattuso has not fielded any questions on the matter recently as the club have imposed a media blackout since February.