It was Liverpool's third successive Champions League away defeat - the club's worst run in 39 years - and Klopp's side will travel to Paris Saint-Germain later this month before hosting Napoli at Anfield on December 11 in the final round of group matches.
This season Jurgen Klopp's side's priority is the Premier League, where they've started brilliantly, but they still have designs on trying to better their surprising run from last season.
Red Star are expected to be without Vujadin Savic due to injury but Marko Marin is set to return to the squad.
They got a hard group stage draw when PSG and Napoli came out of the hat with them but they continued their excellent European form with an opening round win against the Ligue 1 champions to put themselves in pole position.
As it happens, Liverpool can afford to draw in Paris and still qualify from the group and you could see them beating many of Europe's top sides over two legs, as they showed a year ago.
Liverpool finishes the group with a trip to PSG and a home game against Napoli. Napoli found it tough when they came here and we knew it was going to be a tough place to come.
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"It's on us to take responsibility and we need to bounce back in the next two games".
'We didn't start well enough, ' Milner said.
The second half was little better for Liverpool and it took until the 70th minute before they were able to truly test Red Star's keeper, Milan Borjan. We had moments, we had kind of nearly-chances, the biggest in the last two three minutes and before that not enough. They're probably the only ones this season that we haven't, for whatever reason.
With the confidence flowing, Pavkov then doubled Red Star's advantage six minutes later, holding off the challenge of Georginio Wijnaldum before steadying himself and firing home from 25-yards out.
The prolific front three of Mohamed Salah, Roberto Firmino and Sadio Mane, who combined for 91 goals last season, have all suffered a downturn in form at different stages through the first three months of the campaign.
Liverpool had the briefest of openings, but, uncharacteristically, Adam Lallana's first touch let him down as he tried to control Salah's square pass inside the penalty area.
They conceded a goal from a corner and another from a loss of possession and quick counter-attack against Red Star.