Having threatened all night without anything to show for it besides Dries Mertens' volley off the crossbar, Napoli finally broke through when Insigne redirected a cross from Jose Callejon by the far post.
Insigne, who grew up in the Naples area, celebrated by creating a heart shape with his hands.
"It's wonderful to win like this," Insigne said. "We had a great performance as a team and wanted the result at all costs."
The victory vaulted Napoli atop Group C with four points, one ahead of Liverpool and Paris Saint-Germain, which beat Red Star Belgrade 6-1 in the other group match Wednesday. Red Star, which held Napoli to a scoreless draw in the opening round of matches, is last with one point.
"We know it's a difficult group," Insigne said. "But we want to make our mark in every game and put any opponent under pressure."
With Liverpool midfielder Naby Keita taken off injured on a stretcher early in the first half, the visitors were restrained and Napoli defender Kalidou Koulibaly didn't allow Mohamed Salah, Liverpool's most dangerous striker, any room at all.
Liverpool said in a statement on its website that Keita was taken to a local hospital where he "underwent a number of precautionary health checks to assess his condition." The team, which returns to England on Thursday morning, said no decision had been made on when Keita would be discharged.
With the likes of Salah, Roberto Firmino and Sadio Mane — the trio whose goals propelled Liverpool to last season's Champions League final — the Reds had scored 41 goals during their last 12 Champions League matches and had only failed to score in one of their last 18 games in the competition, including qualifying.
Against Napoli, Liverpool failed to muster a single shot on target in a Champions League game for the first time since February 2006 in a 1-0 loss to Benfica.
For Napoli, which had wasted an early lead in a 3-1 loss to seven-time defending Serie A champion Juventus at the weekend, it was the best performance under new coach Carlo Ancelotti.
"The team did well throughout the game. We never lost control," Ancelotti said. "We were always very focused, above all when defending and allowed Liverpool practically nothing.
"I see the quality this team has, but with a little tension we don't express ourselves the way we should," Ancelotti added. "I hope this victory can build confidence and make us play with more freedom."
Napoli supporters chanted long after the final whistle as their team went over and saluted the club's hard-core "ultra" fans, who maintained a drum roll for the entire match amid a deafening atmosphere.
After seven straight wins in all competitions to open the season, Liverpool has now gone three matches without a victory, having lost to and drawn with Chelsea in its previous two fixtures.
One Liverpool fan was injured in clashes with Napoli supporters the night before the match. There was also a flare thrown into the area where Liverpool supporters sat inside the stadium.