Dries Mertens scored from the penalty spot with eight minutes remaining after Andy Robertson tripped Jose Callejon inside the box and Napoli substitute Fernando Llorente added a second in stoppage time to put the result beyond doubt, Reuters reported.
Klopp’s side became the first competition winner to lose the opening game of its Champions League defense since AC Milan was defeated by Ajax Amsterdam in 1994 and the German was concerned by his players’ wastefulness.
“It should hurt because there were opportunities for us,” he told BT Sport.
“It was an open game with a lot of counter attacks, but we didn’t finish them off and that is a problem. In the second half it was a wild game, they were running and we were running.”
Klopp was also unhappy about the penalty awarded to the home side after Robertson was adjudged to have fouled Callejon.
“I don’t think it’s a penalty,” he said. “What can I say, for me, it is clear and obvious no penalty. He jumps before any contact, we can’t change that.”
Carlo Ancelotti insisted that there was no sense of "euphoria" after the victory over Liverpool.
"Euphoria at what? We won the first match, it was a good test and we come out of this match aware that this team can do well," said Ancelotti, AFP reported.
"It was just one victory, but yes it has given us confidence, being able to match Liverpool is impressive as for me they are the best team in Europe at the moment.
"It's proven that this team can perform very well, dig in and make sacrifices."
Ancelotti has started his second season at Napoli with the Italians going out in the group stage last year after losing to Liverpool at Anfield, having beaten Klopp's side beforehand in Naples.
"We brought in important players who improved the quality of the squad and have different characteristics to what we had before," said the 60-year-old, whose side is fourth in Serie A.
The three-time Champions League winning-coach – twice with AC Milan and once with Real Madrid – said his focus was just on getting out of the group stage.
"I thank (Liverpool coach Jurgen) Klopp for saying we can win this competition," said the Italian, "but it's a long way to go. Now we must think about progressing to the knockout stages."
Belgian Mertens added, "We beat Liverpool a year ago as well, it counts for nothing if we don’t do well in the next matches."
Napoli next plays Belgian side Genk on October 2, followed by a trip to Salzburg, before visiting Anfield on November 27.
In the other Group E game, Salzburg made an impressive start with a 6-2 victory at home to Genk.
Erling Haaland scored a first-half hat-trick for the Austrian champion, becoming the third youngest player to net a Champions League treble at the age of 19 years and 58 days, behind Raul (18y 113d) and Wayne Rooney (18y 340d).