Napoli's first opportunity of the match arrived in the 11th minute when Simone Verdi worked the ball into a shooting position, but Simon Mignolet was on hand to make the save. The stopper then kept out a free kick from Dries Mertens during a decent spell for the Italian side.
The Serie A outfit continued to threaten as the match developed and they made the breakthrough in the 17th minute when Insigne picked out the bottom corner with a fine finish after a mazy run.
Napoli were full of it, though, and scored their second in the 29th minute when Milik converted a brilliant low cross from the dangerous Insigne.
The movement of Insigne, Mertens and Jose Callejon was proving to be too much for the Reds, who needed Virgil van Dijk to make a couple of important interceptions in order to keep the score down in the first half.
Klopp's side did have the ball in the back of the Napoli net late in the first period courtesy of a fine finish from Wijnaldum, but the offside flag cut the English side's celebrations short.
Napoli scored their third of the match in the 52nd minute, though, as Younes - one of three changes at the interval - converted from close range after Mignolet had palmed away a strike from Insigne.
Mignolet had to keep out an effort from Milik in the 58th minute as the Italian outfit chased more goals against a Liverpool side that were second best by some distance in Scotland.
Not too much of note occurred in the period that followed as the stream of changes made it difficult for either side to put together a spell of attacking football. There was a notable introduction off the Liverpool bench in the 79th minute, though, with recent arrival Harvey Elliott taking to the field.
Liverpool came close to scoring late on when a cross from Brewster was met by Napoli defender Elseid Hysaj, but Meret was on hand to make an excellent save before the goalkeeper made an even better stop to keep out Wilson's drive late on as Ancelotti's side kept their clean sheet intact.
Next up for Liverpool is a clash with Lyon on Wednesday night ahead of the Community Shield against Manchester City next weekend. Napoli, meanwhile, will continue their preparations for the new season against Marseille on August 4.
LIVERPOOL: Mignolet; Alexander-Arnold (Hoever 73'), Matip (Lovren 62'), Van Dijk (Van den Berg 79'), Robertson (Gomez 73'); Wijnaldum (Elliott 79'), Fabinho (Duncan 79'), Henderson (Lallana 62'), Milner (Lewis 79'), Oxlade-Chamberlain (Wilson 62'); Origi (Brewster 62')
NAPOLI: Meret; Di Lorenzo (Hysaj 61'), Maksimovic (Luperto 61'), Manolas (Chiriches 46'), Mario Rui (Younes 46'); Callejon, Zielinski, Verdi (Gaetano 61'), Mertens (Ghoulam 46'); Insigne, Milik (Tutino 75')