Jurgen Klopp’s side did not manage a shot on target in the first half, but were ahead early in the second after Pascal Gross fouled Mohamed Salah in the area and the Egyptian fired in the spot-kick.
Victory extended Liverpool’s advantage at the summit ahead of third-placed Tottenham’s match with Manchester United on Sunday and second-placed Manchester City’s meeting with Wolves on Monday.
Burnley 2-1 Fulham
Andre Schurrle fired Fulham ahead just two minutes in with a stunning strike which crashed in off the underside of the crossbar.
But, just after Chris Wood had struck a post for the hosts, they were level in the 20th minute when Jeff Hendrick’s shot deflected off Joe Bryan to find the net.
And three minutes later they were ahead as another Hendrick shot this time deflected off Denis Odoi to beat Sergio Rico.
Cardiff City 0-0 Huddersfield Town
It finished goalless between Cardiff and Huddersfield, with Huddersfieldangered late on when referee Lee Mason initially awarded a penalty for a foul on Florent Hadergjonaj, only to change his mind after talking to his assistant.
Crystal Palace 1-2 Watford
Watford struck a post twice in the opening five minutes against Crystal Palaceand their failure to score looked like costing them dear seven minutes before the break when a goalmouth scramble ended with a Craig Cathcart own goal.
But Cathcart made amends in the 67th minute when he scored at the other end to level and seven minutes later Tom Cleverley smashed a loose ball into the top corner to set Watford on their way to a 2-1 victory.
Leicester City 1-2 Southampton
James Ward-Prowse put the Saints ahead from the penalty spot in the 11th minute after Shane Long was brought down by Nampalys Mendy.
The Foxes were handed a lifeline when Southampton defender Yan Valery was sent off shortly before the break, collecting his second yellow card after jumping in on Mark Albrighton.
But in first-half stoppage time Long made it 2-0 to Saints after pressuring Mendy and firing beyond Kasper Schmeichel.
Wilfried Ndidi got Leicester back in it just before the hour mark, bundling home from close range, but Southampton saw the game out.