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.
Mohamed Salah fired over the bar shortly afterwards to sum up a poor first half for Liverpool, with Daniel Sturridge also missing an early chance for the visitors when he volleyed over the bar from close range.
Pakov and Red Star celebrate.
"Napoli came here and found it tough as well, so it's a tough place to come, but we expected that".
"I saw, unfortunately, a few games like this already and it's really hard to find your mojo back", surmised Klopp as Belgrade rocked to Red Star's first win in the competition for 26 years.
And it has ultimately left Liverpool with real work to do in their final games away to PSG and at home to Napoli. We had moments, we had kind of nearly-chances, the biggest in the last two three minutes and before that not enough.
Liverpool started the game at the top of Group C and knew victory would move them to the brink of qualification for the last 16.
The over that went for 43 runs!
The stand was a Northern Districts record for the sixth wicket, surpassing the 150 set by BJ Watling and Peter McGlashan. South Africa-born fast bowler Willem Ludick was at the receiving end as he was taken to the cleaners, literally.
"We played well and managed to shut them out".
Pavkov, a surprise starter up front, gave Red Star a 22nd-minute lead when he headed home from the edge of the six-yard box after a Marko Marin corner from the right. "We believe we can win every game, that's always been the case with us".
"Of course, we have to", he said.
"It's on us to take responsibility and we need to bounce back in the next two games".
After a frustrating opening 15 minutes, Liverpool should have been ahead when Daniel Sturridge found himself free at the back post following a unsafe pass across the box, but the striker could only blaze the ball over the bar from just six yards out. Last season we were scoring a lot of goals and they were going in and now they're not.
Red Star scored on 50 percent of its shots on frame, finishing with four shots on goal.
"They time-wasted and we lost momentum but to come in 2-0 down wasn't like us and we've got to take full responsibility and make sure that doesn't happen again".