Andy Robertson and Mohamed Salah hit the woodwork in the second half, as Jurgen Klopp made two half-time substitutions in an attempt to rescue a result.
Last season's runners-up are now top of Group C and one point clear of Napoli at the midway point of the group stage. They arrived in Belgrade in first place in the group but has ceded the initiative and will be overtaken by either Napoli or Paris Saint-Germain, who meet in Naples later Tuesday.
"It was a hard situation if you want, usually you don't think about things like that before a game but we had to think about it".
A victory would extend Liverpool's lead atop to the group standings, and a tie in the other Group C game - Napoli versus PSG - would leave Liverpool in an nearly unassailable position to progress with just two games remaining.
"You can't give teams that start", Liverpool midfielder James Milner said.
They got a hard group stage draw when PSG and Napoli came out of the hat with them but they continued their excellent European form with an opening round win against the Ligue 1 champions to put themselves in pole position. "He's our player, he'll play a lot of games for us but not tomorrow".
Meanwhile, Monaco were handed a humiliating defeat that all but ended their Champions League campaign as Thierry Henry once again failed to secure a maiden win as coach.