MO Salah reached 50 goals for Liverpool as Jurgen Klopp's side hit the front in their Champions League group.
Jurgen Klopp's side swept aside Red Star Belgrade 4-0 at Anfield to take full advantage of the 2-2 draw between Paris St Germain and Napoli in the other Group C game.
Roberto Firmino began a routine night for the Reds in the 20th minute before Salah took over.
The Egyptian netted his first just before half-time and then scored a penalty shortly after the break to make it 50 goals in just 65 Liverpool appearances.
Liverpool could even afford the luxury of missing a second penalty by Sadio Mane.
But the Senegal forward made amends moments later to put further gloss on a second win in three Champions League games for the Merseyside outfit.
Napoli were denied top spot by Angel Di Maria's stunning stoppage-time equaliser for PSG.
Lorenzo Insigne and Dries Mertens had twice given Napoli the lead and the Italians seemed on course to build on their victory over Liverpool earlier this month.
But PSG, who had equalised through Mario Rui's own goal, were given hope of making the last-16 when Di Maria dispatched a brilliant 20-yard curler in the third minute of added time.
Hugo Lloris was sent off as Tottenham's hopes of making the knockout stages were dealt a huge blow in a 2-2 draw at PSV Eindhoven.
Spurs recovered from Hirving Lozano's first-half goal to lead through Lucas Moura and Harry Kane.
Davinson Sanchez also saw a perfectly good goal ruled out by a flagging assistant referee and Spurs appeared well set to revive their Group B qualification hopes.
But Lloris suffered a rush of blood - a foul outside his penalty area on Lozano - earning him a straight red card and, shortly after, former Newcastle loanee Luuk de Jong equalised to leave both PSV and Spurs with one point from three games.
Barcelona lead the way in Group B after beating Inter Milan 2-0 at the Camp Nou.
Even without the injured Lionel Messi, who was watching in the stands, Barcelona proved too strong for their Italian visitors who had won their opening two games.
Rafinha volleyed home just after the half-hour mark before Jordi Alba profited from Ivan Rakitic's vision seven minutes from time.
Thierry Henry's first taste of management in the Champions League saw Monaco draw 1-1 at Club Brugge.
Moussa Sylla put Monaco ahead in the 31st minute to become the second-youngest player to score for them in the competition after Kylian Mbappe.
But Wesley Moraes equalised eight minutes later to give the Belgians a point.
Monaco and Club Brugge are playing catch-up in Group A and Borussia Dortmund took control of the section with a 4-0 thumping of Atletico Madrid.
England new boy Jadon Sancho was among the scorers, with Raphael Guerreiro (two) and Axel Witsel also on target as the German side maintained their 100 per cent record.
In Group D, Porto won 3-1 at 10-man Lokomotiv Moscow to open a two-point lead over Schalke.
Moussa Marega, Hector Herrera and Jesus Manuel Corona scored for Porto while Lokomotiv - who had Solomon Kvirkvelia sent off late on - replied through Anton Miranchuk.
Galatasary and Schalke fought out a goalless draw in Istanbul.
