Milan Pavkov’s first-half double humbled the Reds and blew Group C wide open.
It means Liverpool have now lost three successive Champions League away games for the first time in their history.
Asked to explain this flop, boss Klopp said: “I’ve only ten fingers. There were a lot of moments when we could get in control of the game and didn’t do it.
“We made the wrong decisions. It just happened.
“We will talk about it but we can’t change it.
“We gave them too many set-pieces. They scored with a header after the corner.”
Pavkov’s second was a long-range screamer and Klopp said: “The second goal was out of the blue.
“We gave the ball away and that’s not cool in a game like this, an atmosphere like this, a situation like this.
“You then have to play your best football in a negative mood. It was a blow.
“We tried to change in the second half, we had some moments in their box, scruffy situations.
“Red Star deserved it. After scoring, they were on their bikes and we had to deal with the situation.
“It is not what we wanted and we will go for the other two games 100 per cent.”
Liverpool face Paris Saint-Germain away in three weeks before the final home game against Napoli.
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Midfielder James Milner said: “We didn’t start well enough and that showed by conceding those two goals.
“We did get better in the second half — we had enough chances and didn’t take them. And when that happens you get punished.
“We had problems all over the field. We didn’t win it back well enough. Our passing wasn’t good enough and we didn’t start at all.
“We kept fighting and kept playing, which is positive, but you can’t give them a start in this competition.
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“We need to win our games now.
“It’s on our shoulders and as long as we can do that we will be fine. We know we can beat them. But we have to play better than tonight.”
Red Star coach Vladan Milojevic said: “I dedicate this win to our supporters. They were phenomenal.
“It is very important for the whole country.”
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