The Uruguayan hitman believes the Partenopei needed to get a result against their Ukrainian visitors and was delighted to help his side with a man-of-the-match performance
The Uruguayan struck four times for the Italians, with three of his goals coming in the last 13 minutes of play to ensure a 4-2 win on the night.
Cavani's goals have helped Napoli move up to second place in Group F, leaving them in a good position to progress to the next round.
"We have been in a delicate moment lately. We had to send a strong signal with our performance tonight," he told reporters after the match in Naples.
"Winning tonight was important and we succeeded. I'm very happy with my efforts. Tonight I scored all the goals and I gave 100 per cent to help contribute to my team. It was really a wonderful evening. The goals are for my wife and my family."
Despite being his side's saviour on Thursday, Napoli boss Walter Mazzarri has insisted that his team do not solely rely on the 25-year-old.
“I have a team of champions, not just Cavani. I am proud of all these lads,” he told reporters.
“It seemed to be another cursed night for us, but thanks to the group we were able to turn it around.
“I give compliments to all my players tonight and not just to Cavani. Anything can be done, we just have to work on keeping the balance."
Mazzarri was a key figure in bringing the Uruguay international to the San Paolo in 2010, but, also revealed that he used the help of his national team-mate, Walter Gargano, to convince him to come to Napoli.
“We wanted Cavani in 2010, I believed in him and hoped he could become the kind of top player that he effectively is.
"When he arrived, the Napoli fans weren't too happy, as Fabio Quagliarella was leaving.“I convinced him with the help of Walter Gargano that Napoli was the right club for him. He exploded here because the team plays the right kind of style for him."