Jurgen Klopp decided to make some changes from the side that drew at Arsenal last weekend with Lallana coming in for Fabinho in midfield while Daniel Sturridge was also preferred ahead of Roberto Firmino up front.
Pavkov broke the deadlock 22 minutes in with a towering header from Marko Marin's corner after evading the attentions of Virgil van Dijk.
Milan Pavkov scored a first half brace for Red Star Belgrade.
Forward Milan Pavkov headed his side into the lead after 21 minutes and fired a powerful effort into the bottom corner to make it 2-0 shortly afterwards.
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The caravan, estimated at about 3,500 people, is now hundreds of miles away and would take at least six weeks to reach the border. These include putting up more fences, creating temporary vehicle barriers and operating heavy machinery.
'Napoli found it tough when they came here and we knew it was going to be a tough place to come. The second goal was kind of out of the blue; we had the ball and gave it away.
Crvena Zvezda fans had a choice greeting for Liverpool when the two sides met in Belgrade for their UEFA Champions League Group C clash on Tuesday - with a pre-match choregraphed rendition of "F*ck you Liverpool".
There were plenty of candidates for worst player, but Daniel Sturridge's terrible first half performance was noticeably awful.
"It just happened. We will talk about it but tonight we can't change it". I said earlier that I'd like to score against Liverpool and it came true.
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Last month, Facebook removed 82 pages , accounts and groups tied to Iran and aimed at stirring up strife in the USA and the U.K. It carried out an even broader sweep in August, removing 652 pages, groups and accounts linked to Russian Federation and Iran.
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. At half-time we walked a long way to the dressing room and they were all in a very positive mood. Tonight when we saw he would play, for us it was clear it would be in a specific way different because he is a proper target player for Red Star.
Liverpool manager Jurgen Klopp said afterwards: "The boys are very disappointed, I'm very disappointed and we have to do better".
Liverpool's terrible display in their defeat to Red Star Belgrade on Tuesday night can be summed up in one statistic - zero succesful tackles in midfield. 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.
But Red Star were worthy of the victory - the first they have claimed in the Champions League for 26 years - in front of a noisy and passionate home crowd in the Rajko Mitic Stadium.
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He added that police were dealing with an "active situation", meaning it is not known if the attacker has been subdued. A gunman opened fire at a bar in Thousand Oaks, California on Wednesday evening, ABC News reported.