"We had the first big chance with Daniel [Sturridge], a very big one. It's not what we wanted", Klopp told BT Sport. "One of them, they scored and the block worked [on] Virgil in that area".
Van Dijk was easily outjumped for the opener by Pavkov before giving the same man far too much room to shoot from distance for his second shortly after. They were, if you want, on their bikes and we had to deal with the situation.
This will be Liverpool's first away game against Red Star Belgrade in European competition since October 1973, when they lost 2-1 in the first leg of a last-16 tie in the European Cup.
Klopp opted to rest Roberto Firmino and give Sturridge a rare start, but he endured a hard game.
He leapt highest at a corner to head in a right-wing delivery from Marko Marin and put Red Star ahead.
Republic of Ireland underage worldwide Caoimhin Kelleher has been called up to the Liverpool squad to travel to Serbia to face Red Star Belgrade on Tuesday.
Changes were needed and Klopp acted swiftly, bringing on Roberto Firmino and Joe Gomez at half-time for Sturridge and Trent Alexander-Arnold.
Sadio Mane had a great chance to give the visitors a lifeline, but failed to force the ball across the line from point-blank range.
There were a whole host of Liverpool players at fault for Red Star's second.
However, they are a different proposition at home and after holding Napoli to a 0-0 draw in their group opener, they kept Liverpool at bay after the break.
Klopp isn't anxious about the safety of Shaqiri, rather he'd prefer to keep the narrative around football than politics.
Kosovo is a former Serbian province that declared independence in 2008, although Serbia does not recognise this.
The Serbian champions will hope that their supporters will make a difference on Tuesday evening and help them get a positive result to make some noise in a highly competitive Champions League Group C which also consists of Italian power Napoli and French champions PSG.