Mo Salah scored the only goal of the game from the penalty spot as Jurgen Klopp's side went seven points clear at the top.
That could of course change when Manchester City host Wolves on Monday night.
Elsewhere, Arsenal's top four hopes were hit with a 1-0 away defeat to West Ham, wonderkid Declan Rice bagging the winner.
They stay fifth – six points behind fourth-placed Chelsea who secured a 2-1 win at home to Newcastle in the evening kick-off.
Burnley beat Fulham despite not having a shot on target while Leicester crashed 2-1 at home to ten-man Southampton.
Watford defeated Crystal Palace 2-1 away from home while Cardiff and bottom-club Huddersfield drew 0-0.
Catch up on all the highlights from this weekend's games so far, below…
West Ham 1 Arsenal 0
Brighton 0 Liverpool 1
Burnley 2 Fulham 1
Cardiff 0 Huddersfield 0
Crystal Palace 1 Watford 2
Leicester 1 Southampton 2
Chelsea 2 Newcastle 1
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