Marko Marin delivered from the right-hand side, and Pavkov - who had never before scored in the Champions League - rose unmarked to nod into the bottom corner on 22 minutes.
The Reds will visit Red Star Belgrade on Tuesday in a Champions League Group C game.
"The boys are very disappointed, I'm very disappointed and we have to do better".
The Reds now occupy top spot in their Champions League group and will want to replicate their 4-0 result in the last meeting between the two sides.
"It's a situation where we will go to the fantastic city of Belgrade as a football team to play football, Klopp told Liverpool's website". We all have to deal with it.
"That's why we try to make sure that we can be focussed on football".
"Then more or less straight away they score the second one and then the atmosphere was completely different, you could feel it".
The Serb headed the hosts in front in the 22nd minute before striking a stunning 25-yard drive past Alisson after Liverpool failed to close him down just seven minutes later.
Despite Liverpool hitting the woodwork twice, Red Star held on for what was a deserved victory.
When Red Star travelled to Anfield last month, Shaqiri was roundly jeered by the away fans, showing that his actions had certainly stayed fresh in the minds of some Serbians.
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.
Ahead of the match, commenting on the prospect of receiving a hostile reception, the midfielder said he "can handle this, no problem", adding that, "It is not about politics, it is about football".
Shaqiri is an ethnic Albanian from Kosovo, the former Serbian province and scene of Nato's military intervention during the conflict between the countries that ended in 1999. The club has also enjoyed a strong start to the Premier League, now sitting in third place, just two points behind leaders Manchester City.