Liverpool were stunned by a first-half brace from Milan Pavkov and could not find a way back into the game despite having the majority of the possession.
Red Star Belgrade comes into Tuesday's match at the bottom of Group C, amassing one point through three games played.
PSG's 1-1 draw against Napoli keeps Liverpool top of the table on six points but only on goal difference from the Serie A side with the French champions a point behind. We have to do better because we can do better, but tonight it is too late. You then have to play your best football in a negative mood.
But Liverpool should have been in front in the 17th minute when the ball ran kindly for Daniel Sturridge inside the six-yard box, but the England global somehow managed to blaze over the bar when completely unmarked.
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Liverpool had a lucky escape at the start of the second period - the referee failing to award Red Star a penalty when a shot struck the arm of James Milner.
There were plenty of candidates for worst player, but Daniel Sturridge's terrible first half performance was noticeably awful.
"We come here as a football team". Very dominant, didn't give a lot of counter-attacks away but we had moments we had "nearly" chances but the biggest chances were in the last two or three minutes. We never came back into situations. A few we had to make and a few we wanted to but it is easy to say it didn't work out in the end. Before that, Liverpool had scored in 14 straight games.
After the Premier League side's routine 4-0 win over the Serbs last month, the former Bayern Munich victor said he could "handle it, no problem", in reference to playing in Serbia.
"We are dedicating this game to our supporters because they were phenomenal". Unfortunately, they could not get into the game and eventually lost 2-0 which made their Champions League campaign even harder.