After ending Napoli's perfect record with a victory at home, the Reds had to settle for second place in the group.
The Reds were looking to bounce back from Saturday's home defeat by Leeds United and their spirits will be lifted by late goals from Mohamed Salah and Darwin Nunez.
The three points will act as a real boost to the spirits of the players, despite the fact that they never looked like securing the four-goal winning margin.
In the first half, Thiago brought a save from Alex Meret, while in the second half, Napoli thought they had taken the lead, only for the goal to be ruled out for an obstruction.
A game of few chances looked to be heading for a draw, but with five minutes to go, a close-range finish by Mohamed Salah gave the Reds the victory.
One of the features of their disappointing start to the season has been the vulnerability of their defence and Ibrahima Konate showed what they have been missing.
Konate suffered a pre-season knee injury that forced him to miss the start of the season, then he was out for a second time due to a muscle problem that left his backguard struggling.
The France defender returned against the formidable Napoli attack and gave an outstanding display as the Reds looked a lot more secure than they have been recently.
Konate's defensive quality alongside Van Dijk kept Victor Osimhen at bay, showing composure, pace and physical presence as Napoli suffered the rare taste of defeat.
Even though Napoli were not at their most destructive, the class demonstrated by Konate illustrated exactly why manager Juergen Klopp will hope he can have an injury-free run for the rest of the season.
The team that spent nearly all of last season in pursuit of a historic quadruple has looked a shadow of its former self after defeats by both Forest and Leeds.
The context is that the Italians knew only a four- goal margin of defeat would make a difference in the group.
It removed a large part of jeopardy from the group, and there was never any chance of a frenzy of expectation for the team.
The two faces of their season so far have been shown by this example. They have lost to teams who have spent the early part of the season in struggle, as well as to teams they have punished in the past, such as Manchester United andArsenal.
The Reds looked more secure at the back, survived a VAR scare when Odigard thought he had scored, then ended the match on a high with those two late scrambled goals that will act as a real mood change for the team and the fans who have been so disappointed.
When they travel to Spurs on Sunday, they will face a serious test of their credentials as they look to get back to winning ways after a poor start to the season.