Red Star Belgrade were always going to find this a difficult test, with a trip to Anfield one of the most daunting prospects in European football.
Liverpool managed to keep hold of the ball well in the early minutes, but were not quite able to find the final ball to help them get a foothold in the game.
All that changed in the 20th minute, thanks to Xherdan Shaqiri´s brilliant pass to feed in left-back Andy Robertson on the flank. The Scotsman managed to pick out Roberto Firmino in the centre of the box, who got the ball out of his feet and fired a deflected shot in to the goal.
That goal meant that Liverpool were able to dictate the rest of the first period, combining patient build-up play and high-pressing to keep Red Star's possession stats to just 38 % in the first half.
Jurgen Klopp's men managed to confirm their dominance by notching a second goal just before the half-time whistle, with Shaqiri's brilliance again on display.
Georginio Wijnaldum fired the ball in to the former Stoke man on the edge of the box, who with a brilliant flick managed to get it to Mohamed Salah. The Egyptian was able to put his poor goal-scoring record behind him as he fired the ball high in to the goal and make Red Star's task in the second half a seemingly impossible one.
It was not long before Liverpool made it three, as Sadio Mane was judged to have been brought down in the box by a flailing arm from the Red Star captain. Salah was confident enough to step up to double his tally for the night, and he did just that with as he hammered the ball down the centre of the goal.
Red Star found it more and more difficult to get the ball of their opponents as the match progressed and their legs grew heavier, not forcing Alisson to make a save in the match despite almost grabbing one back through a sliding Boakye who couldn't get on the end of a drilled cross.
The referee gave another penalty on 75 minutes for a somewhat debatable hand ball, but Sadio Mane was not able to convert as he drew a fine save from goalkeeper Borjan.
Fortunately for the Senegalese he was able to atone for his miss just minutes later, when Danny Sturridge found him unmarked at the back post and the winger prodding the ball home to make it 4-0.
Jurgen Klopp will be thrilled with the way that his men kept pushing for more goals right up to the final whistle, as well as being able to keep a clean sheet at the other end, as they were able to capitalise on PSG's loss in the other match in their group.
Red Star Belgrade put in a valiant effort, and they avoided it being more, but were ultimately out-classed by a dominant and relentless Liverpool side.