The side from the Serbian capital had actually had the better of the opening exchanges, in front of a packed house at the Rajko Mitic stadium, and they were good value when they went in front.
After an earlier penalty shout was turned down, a corner from the right was swung in and Pavkov got the better of his marker to send a towering header into the far corner.
Liverpool were warned before the game that the atmosphere in eastern Europe could be intimidating, and they gave the home crowd every reason to make some noise when they concede. The roof was lifted.
It is a goal that could have far-reaching ramifications in a very competitive Champions League group, as Napoli and PSG are both waiting to pounce on any Liverpool shortcomings.