The image of the tearful fan with a Napoli scarf appeared during the broadcast of the match and was widely posted on social media.
It prompted some supporters to suggest it should be hung on the team’s dressing room wall to motivate the team.
Napoli, runners-up in three of the last four seasons, have slumped to 11th in Serie A this term.
They have lost four of their five league matches since Gennaro Gattuso replaced Carlo Ancelotti as coach in December.
Gattuso described Saturday’s performance as embarrassing.
“I’m sorry that the result made you cry,” Insigne said in a video to the fan, named as Mario, which was posted on several Italian websites.
“We promise that from now until the end we will have a great championship and you will smile with us again. We are waiting for you at the stadium. A big hug.”(Reuters/NAN)