Carlo Ancelotti's side dropped into the Europa League as Group C's third-place finisher.
PSG topped Group C with 11 points. Despite tying Napoli on nine points, Liverpool advanced due to a superior number of goals scored in all matches.
Mbappe gave a magnificent performance in Belgrade, figuring into three of PSG's goals to keep his side alive in a competition they're expected to win.
The match between Liverpool and Napoli was much closer. It started with one of the most contentious calls in the group stage, defender Virgil van Dijk catching Dries Mertens in the ankle with a studs-up challenge. The Dutchman only received a yellow card for what was widely panned as a red-card offense.
Sadio Mane botched several chances to double Liverpool's advantage, and the hosts were nearly made to pay for his wastefulness. If not for Alisson's miraculous last-minute save on Arkadiusz Milik, the Serie A outfit would have claimed a 1-1 draw and progressed with a superior head-to-head record.
"I'm still full of adrenaline," Klopp told reporters. "This game was just amazing. The boys played with their whole heart. Our defending and pressing was some of the best I ever saw.
"The most difficult period was right after the 1-0. It was really wild from both sides. What a goal from Mo, and the save (Alisson) made, I have no words for that. I am very proud of the boys tonight."