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.
Milan Pavkov scored a first half brace for Red Star Belgrade.
The home side made a fast start and deservedly picked up their first win in the pool courtesy of Milan Pavkov's first-half double. They remain unbeaten in the Premier League but have rarely reached the heights that made them so scintillating to watch last term.
Red Star, who are playing in the competition's present format for the first time since they won the European Cup in 1991, also have four points after a spirited performance.
The 24-year-old forward was thrilled with the success, which has suddenly boosted Red Star's chances in Group C, and said it would be tough to put behind him quickly. We had the first big chance - Daniel [Sturridge] with a very big one. If - a big if - you score early in the second half the game can change, but it didn't.
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The side was hamstrung by the inability of that threesome to win the ball - they also conspired to lose it on a number of occasions, notably Milner for the second Red Star goal. At half-time we walked a long way to the dressing room and they were all in a very positive mood.
They didn't give up fighting, they stayed concentrated and they gave everything.
While there may be some excuse in the fact that Liverpool dominated possession - 63 per cent - it is factors such as an inability to stick their tackles which infuriated manager Klopp after the game. [There were] a lot of balls which don't count as a shot on target because we missed the target. They deserved it for the passion they showed. However, defeat means they can not really afford to lose either of their two final round-robin matches against PSG and Napoli, who are both impressive sides.
If Liverpool draw with PSG and Napoli draw with Red Star, then Liverpool will need to beat Napoli in order to go through, or draw with them and hope that Red Star get a result against PSG.
The visitors began to mount some real pressure on the Red Star goal during the final exchanges, but they were unable to carve out anything of note in front of goal.