The Premier League outfit's attempts to claw things back saw them hit the frame of the goal twice in the second half through Andrew Robertson and Mohamed Salah, but they were unable to make their pressure tell and Red Star held on to their first win in the group stage.
Liverpool FC have taken six points from their opening three Champions League fixtures and they will be hoping to claim the win on Tuesday night in Serbia to take a step towards reaching the first knockout round of Europe's elite club competition. The boys are very disappointed, I'm very disappointed.
Having demolished Red Star at Anfield a fortnight ago 4-0, it was expected to be a fairly straight-forward three points for Jurgen Klopp's men, who were looking to cement their place at the top of their tough Champions League Group C table which includes Paris Saint-Germain and Napoli.
And Milner, speaking to BT Sport after the game, bemoaned a sloppy start, missed opportunities in front of goal and an inability to counter the fierce atmosphere.
Liverpool finishes the group with a trip to PSG and a home game against Napoli.
Milan Pavkov scores a spectacular second goal.
"They had too many set pieces that's true and with one of them they scored", added Klopp.
Pochettino: Spurs will realise their dream
Speaking at his press conference, van Bommel said ahead of the tie, "We did not do well at home, we have to be honest about that". Mauricio Pochettino has promised to stay at Tottenham until he restores the club to the dominant force they were in the 1960s.
The fearsome Liverpool trio is relishing the chance to silence hostile fans at Red Star's Rajko Mitic Stadium on this European adventure.
Klopp opted to rest Roberto Firmino and give Sturridge a rare start, but he endured a hard game.
Klopp responded at half-time by introducing Roberto Firmino and Joe Gomez, although Liverpool still failed to break through a resolute opponent.
The outstanding Borjan again denied Salah from close range in the 82nd minute as Liverpool piled on the pressure but the visitors could not find a way through Red Star's packed and well-organised defence. Liverpool try to turn home the rebound but it won't quite drop for them and Red Star hack it to safety.
Meanwhile, Klopp also discussed his decision to leave Xherdan Shaqiri at home.
Shaqiri comes from an ethnic Albanian family from Kosovo, a former Serbian province.
Left-back Andy Robertson, playing nearly as an out-and-out winger after the break, saw a cross deflected on to the top of the crossbar, while Salah finally forced a save out of Milan Borjan, hitting a post with his next effort.