Failure to beat PSG in Paris would set up a tense must-win finale which could also entail needing a win by two clear goals to redress their head-to-head record against the Serie A side.
Goalkeeper Alisson Becker was forced to save low to his left after El Fardou Ben Nabouhane's shot from distance, but, from former Chelsea midfielder Marko Marin's resulting corner, Pavkov headed home.
"We are keen to be respectful and keen to avoid any distractions that would take focus away from a 90-minute-plus contest that is important for football and only football".
Milan Pavkov scored in the 22nd and 29th minutes as the Serbian champions earned a first win in their debut appearance in the Champions League group stage. We made life a bit too easy tonight for Belgrade, that's the main problem.
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It may be a little harsh as Lallana was excellent for Liverpool during Klopp's early years at the club but a succession of injuries have taken their toll and it remains to be seen whether the England global will be able to return to his best form. But the atmosphere after the second goal especially, in the whole club, if you want, they could now smell it. We have a long flight home and have enough time to think about it.
They didn't give up fighting, they stayed concentrated and they gave everything.
"You could see that a few things were not going too well", Liverpool manager Jurgen Klopp told BT Sport. [There were] a lot of balls which don't count as a shot on target because we missed the target.
That was of little concern to Red Star's players, however, as they joined arms and took the acclaim of their supporters on the full-time whistle.
And Red Star doubled their lead just seven minutes later when Pavkov bagged his second with a superb strike from 25 yards out which Alisson may feel he could have done better with. They deserved it for the passion they showed.
The pool is now wide open and with a hard trip to PSG followed by the visit of Napoli, the Reds need to up their level quickly if they are to navigate a path into the knockout stage.