Ancelotti was sacked just hours after taking Napoli into the Champions League last 16 with a 4-0 win over Genk.
Gattuso, 41, previously managed AC Milan but left the Serie A club at the end of last season.
“We need to work hard, win back points and come through this tricky period,” said Gattuso.
“The aim is to recoup points and get back into the Champions League positions.”
Napoli are seventh in Serie A, eight points off the top four.
Gattuso played under Ancelotti at AC Milan and said: “He is like a father to me. He’s always been close to me and supported me. He’s won it all and I still have a lot to prove.”