A 1-1 draw between Napoli and Paris Saint-Germain in Group C's other game on Tuesday means just two points separate the four teams heading into the final two matchdays. Klopp said the team "lost its mojo" in Serbia.
"We had the first big chance with Daniel (Sturridge), a very big one".
"I honestly hope this guy is released by us in January because he simply doesn't deserve to be here anymore".
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.
"From the first whistle, we didn't start well enough, and that showed with them getting the two goals".
Liverpool tried to force their way back into the game and came close through Robertson and Salah.
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Liverpool ideally need to get a result with a tricky tie away in the French capital to come next before a potential qualification decider at home against Napoli at the end of the group.
Red Star are bottom of the group on just one point and have virtually no chance of qualifying for the knockout stages ahead of Liverpool, Napoli or PSG so coach Vladan Milojevic has admitted they are playing for pride this evening. Four points from their last two games would be enough to guarantee the Reds' progress to the last 16. Their passing was sloppy, their midfield was ill-disciplined and they seemed to lack any sort of clear game plan.
Adam Lallana nearly got Liverpool back in the game moments later but his header was deflected just wide, and Sturridge then fired wide from just inside the box.
"As players we've got to take full responsibility and make sure it doesn't happen again", Lallana told LFCTV.
The 24-year-old, whose career has stalled badly following his £20 million move to Anfield from Benfica in 2014, was caught liking a tweet celebrating Red Star's victory over Liverpool.