Things turned from bad to worse for the Reds when Pavkov doubled Red Star's lead with a stunning long range strike in the 29th minute.
Whether it handed momentum to the Serbians or not, the home side pushed on from there and were 2-0 up at half time against an unusually flat and unimpressive Liverpool display.
Nenad Krsticicuels tussles for the ball with Mohamed Salah.
He leapt highest at a corner to head in a right-wing delivery from Marko Marin and put Red Star ahead.
The striker soon got his second, shrugging off Georginio Wijnaldum and beating Alisson with a fierce 25-yard drive to leave Klopp looking stunned on the touchline.
The visitors almost got lucky 10 minutes after half-time when Andrew Robertson's deflected cross bounced back off the top of the crossbar.More news: A Utah Town Remembers Its Mayor, Killed In Action In Afghanistan
Liverpool might have dropped two points in the Premier League away to Arsenal but Klopp will be expecting his side to get back to winning ways in the next two fixtures as Red Star and Fulham prepare to face the Merseysiders.
Liverpool, last season's finalist, arrived in Belgrade in first place in the group but will be overtaken by either Napoli or Paris Saint-Germain, who meet in Naples later Tuesday.
If Liverpool beat PSG, and Red Star draw with Napoli then Liverpool can lose to Napoli in the final game and still go through.
Had Liverpool won, they would have moved three points clear at the top of the group and only needed a single point to guarantee their progression to the next round.
"We are Liverpool, a big club, a football team but we don't have any message further than that".
"In the second half I thought we did better, obviously, we had enough chances to get back into the game, but we didn't take them and you get punished".