Milan Pavkov scored twice in seven first-half minutes to hand Red Star their first Champions League victory in 26 years.
Liverpool still lead Group C but could fall behind Napoli or Paris Saint-Germain, who meet in Italy later on Tuesday, and face both those sides in their final two group games.
“The boys are very disappointed, I m very disappointed and we have to do better. We have to do better because we can do better, but tonight it is too late,” said Liverpool boss Jurgen Klopp.
Winners of the European Cup in 1991, Red Star hadn t made it beyond the qualifying stages of the Champions League since the following year.
But roared on by a vociferous home support, the Serbian champions were inspired to condemn Liverpool to a second defeat in the group after also losing at Napoli last month.
“They had too many set pieces that s true and with one of them they scored,” added Klopp.
“Then more or less straight away they score the second one and then the atmosphere was completely different, you could feel it.
“They were on their bikes and we had to deal with the situation.”
Last season s Champions League finalists paid for a sloppy start and Klopp s decision to make three changes from the side that started a 1-1 draw at Arsenal on Saturday.
Klopp handed a rest to Roberto Firmino with Daniel Sturridge starting up front and the English striker blazed over when it seemed easier to score with the goal at his mercy on 18 minutes.
That missed proved very costly as four minutes later Liverpool trailed.
Pavkov rose highest in a crowded penalty area to power home Marko Marin s corner to spark wild scenes of celebration.
Moments later Red Star and their 55,000 fans packed into the Rajko Mitic stadium were in dreamland as Pavkov picked up possession midway inside the Liverpool half, strode forward and unleashed a powerful strike into Alisson Becker s top right-hand corner.
Klopp recognised his initial error at half-time by throwing on Firmino and Joe Gomez at the break.
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.
Red Star had conceded 10 goals in being thrashed by PSG and Liverpool in their last two European encounters.
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.
The decision to leave Xherdan Shaqiri back in Britain due to political tensions will now be questioned as the Swiss international s guile to break down a defence was badly missed.
Kosovan-born Shaqiri caused outrage in Serbia with a pro-Kosovo “double eagle” gesture that represents the Albanian flag during Switzerland s World Cup win over Serbia in Russia.
Klopp s men dominated possession but were largely restricted to efforts from range as Milan Borjan turned Mohamed Salah s shot wide before the Egyptian struck the outside of the post from the resulting corner.
Liverpool are still unbeaten 11 games into the Premier League season, but have now lost their last four Champions League games away from Anfield.