Jose Callejon went down under a challenge from Andy Robertson at Stadio San Paolo on Tuesday, with referee Felix Brych pointing to the spot in the Group E opener.
Replays appeared to indicate contact between Callejon and Robertson was minimal but the decision to give the penalty was ratified by a VAR check.
Dries Mertens squeezed his penalty past Adrian and, after Virgil van Dijk's lapse, Fernando Llorente's first Napoli goal claimed all three points.
Klopp's side lost at Napoli in the group stage a year ago, going on to win the tournament, and the Liverpool manager disagreed with the vital refereeing call.
"I don't think it's a penalty," Klopp told BT Sport. "What can I say, for me, it is clear and obvious no penalty. He [Callejon] jumps before any contact, we can't change that.
"We played a lot of good football but didn't finish it off. We controlled moments but had not enough chances in the end.
"We made decisions that were not right and have to accept the result. It was very often the final ball that was not right."
"I don't think it's a penalty."— Football on BT Sport (@btsportfootball) September 17, 2019
"What can I say? Callejon jumps before there is contact."
Jurgen Klopp accepts the result but he's not happy with the decision that put his side behind 👀
🎙 @DesKellyBTS pic.twitter.com/2HEH9a62Qg
Liverpool are the first Champions League holders to lose the first game of their title defence since AC Milan did so in 1994-95.
"It should hurt because there were opportunities for us," Klopp said.
"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."