However, he insisted it was too early for his team to start thinking about winning the Scudetto.
Cavani, who grabbed Napoli's goal in their 1-1 Champions League draw with Manchester City last Wednesday, steered his team to victory after Alberto Aquilani had given Milan the lead at the Stadio San Paolo.
"I dedicate this hat-trick to God and to my family, who support me and push me. I dedicate it to my wife and my son," the Uruguayan told Gazzetta dello Sport.
"We know what our flaws and our limits are and we know that we must work as we are doing now and do our best."
Asked if this win indicated Napoli had sent out a strong signal that they could be championship contenders this season, Cavani added: "Maybe, but it is a very difficult tournament and on Wednesday we have a complicated game (at Chievo)
"We must continue with this humility. We are happy for the victory and we hope to continue."
Napoli coach Walter Mazzarri added: "We must improve in order to stay at a certain levels. You have to have more personality, must be improved. Maybe we're not used to playing every three days. We were good in defence but I want to see my team be even more authoritarian, we are not completely satisfied with the performance."