Liverpool face the task of qualifying for the knockout round of the Champions League after beating Red Star Belgrade 2-0.
Milan Pavkov scored both goals as the Group C base club scored the competition's first victory, hoping to secure a continuation in Europe.
Liverpool still has Napoli and Paris Saint-Germain to play and now has to make it to the hardest to get into the top two.
But Liverpool should have been ahead in the 17th minute when Daniel Sturridge's ball was friendly in the six-meter box, but the England international somehow managed to jump over the crossbar when it was completely unmarked.
A few minutes later, El Fardou's Ben Nabouhane, Red Star's scorer, delivered a good Alisson parade with a powerful drive from far away. And from the resulting corner, the home side were ahead.
Marko Marin ran from the right side, Pavkov, who had never met in the Champions League, moved into the bottom corner after 22 minutes.
And Red Star doubled his lead just seven minutes later when Pavkov scored his second goal with a superb 25-yard strike that made Alisson feel he could have done better.
Adam Lallana would have got Liverpool back almost instantly, but his header was narrowly deflected, and Sturridge shot straight out of the box.
The visitors were almost lucky ten minutes into the second half when Andrew Robertson's deflected cross bounced off the top of the crossbar.
Liverpool continued to dominate the ball as Red Star sat back. Mohamed Salah took the lead in the 71st minute.
Joel Matip did not manage to get into the box with a high ball and Salah did not manage to buy the loose ball when the time came. Virgil van Dijk climbed the roof from 10 meters away.
Salah had the last injury in injury time when he shot goalkeeper Milan Borjan.
There was no turning back for Liverpool as the Red Star fans celebrated a famous victory.