Mohamed Salah continued his return to scoring form by taking his tally for Liverpool to 50 goals as the five-time European champions eased past Red Star Belgrade 4-0 at Anfield.
The Egyptian ended a four-game goal drought with the winner at Huddersfield on Saturday and struck twice to open his Champions League account for the season after Roberto Firmino had opened the scoring.
Sadio Mane added Liverpool's fourth soon after missing a penalty as Jurgen Klopp's men bounced back from defeat to Napoli in their last Champions League outing to move a point ahead of the Italians at the top of Group C.
Liverpool are now two points clear of free-spending Paris Saint-Germain who needed an injury-time equaliser to draw 2-2 with Napoli in the French capital.
In stark contrast to last season's thrilling run to the Champions League final, Liverpool have struggled for goals in recent weeks with a new-found defensive stability instead keeping them joint top of the Premier League.
Klopp tinkered with his preferred 4-3-3 system by moving Salah centrally as the focal point of a 4-2-3-1 and his decision to hand Xherdan Shaqiri a start paid off as the Swiss added some much-needed creativity in midfield.
After a slow start from the hosts, Shaqiri took control by winning back possession and playing Andy Robertson free down the left.
The Scotland captain's cross picked out Firmino, who cut inside onto his left foot before firing into the roof of the net.
Salah's blushes were spared by a late offside flag after he missed a one-on-one with Milan Borjan after a lovely flick from Mane.
That was the sort of chance Salah has missed plenty of in the early weeks of the season to spark concern that he will fail to match the standards set in an incredible 44-goal debut campaign at Anfield.
However, he is now up to six for the new season as another neat touch from Shaqiri set up Salah to give Liverpool the cushion of a two-goal lead just before half-time.
The home side had luck on their side for their third and Salah's second six minutes after the break.
Mane made the most of an arm across his face from Red Star captain Filip Stojkovic and German referee Daniel Siebert pointed to the spot.
Salah didn't care about the dubiety of the spot-kick as he hammered home the penalty to bring up his half-century quicker than any player in Liverpool's history after just 65 games.
Klopp then withdrew Salah before Liverpool's second penalty of the night to rob him the chance of a hat-trick when Milos Degenek handled inside the area. Mane missed his chance in Salah's absence as Borjan turned his penalty onto the crossbar.
However, the Senegalese made amends 10 minutes from time by stabbing home Daniel Sturridge's pass.