Liverpool were unable to make a come back following two first-half goals from Red Star’s Pavkov.
The reds first big chance of the game in the 17th minute, but Klopp’s men were left to rue Sturridge’s missed effort from close range.
5 minutes later, Red Star broke the deadlock as Marin picked out the head of Pavkov from a corner, who was able to beat Alisson.
Things quickly got worse as Pavkov shrugged off Georginio Wijnaldum less than 10 minutes later and fired a 25-yard thunderbolt straight past the Liverpool keeper.
Despite two half-time changes which saw Daniel Sturridge and Trent Alexander-Arnold subbed off for Roberto Firmino and Joe Gomez, Liverpool were unable to get themselves back into the game.
The Liverpool boss admitted after the game “The boys are very disappointed, I’m very disappointed and we have to do better.
“We have to do better because we can do better, but tonight it is too late.
“It is not what we wanted and we will go for the other two games 100 per cent. We all have to deal with it now.”
It was a brilliant display from the home side, who earned their first victory in the Champions League this season.
A win at Red Star would’ve put the reds in a strong position to qualify from Group C.
Liverpool must now get results away to Paris-Saint-Germain and at home to Napoli to ensure they are to progress into the round of 16, which won’t be an easy task.
Liverpool never do it the easy way, do they?
Man of the match – Milan Pavkov (Red Star Belgrade)
Two goals from the forward saw Red Star Belgrade become the first Serbian side to win a Champions League group game.
Liverpool Player Ratings:
Alisson (5), Trent Alexander Arnold (4), Joel Matip (5), Virgil Van Dijk (6), Andy Robertson (6), Georginio Wijnaldum (4), James Milner (4), Adam Lallana (4), Daniel Sturridge (4), Mohamed Salah (5), Sadio Mane (5)
Subs: Joe Gomez (6), Roberto Firmino (5), Divock Origi (5)
Categories: Champions League