Emre Can's fourth goal of the season ended Burnley's stubborn resistance as Liverpool claimed an unconvincing win to strengthen their bid for a top-four finish.
The Reds were far from their best and fell behind when Ashley Barnes turned home Matthew Lowton's brilliant defence-splitting pass.
Liverpool equalised on the stroke of half-time with their first shot on target when Georginio Wijnaldum poked in at the second attempt.
Can then secured a second-successive victory for Liverpool with a long-range effort into the bottom corner.
Burnley threatened to snatch an equaliser late on but Lowton hooked over from close range.
It was a game of few memorable moments but the win means Jurgen Klopp's side, who remain fourth, are now five points clear of fifth-placed Arsenal.
Burnley, who are yet to win a game away from home in any competition this season, are 12th.
Relive Liverpool's win against Burnley
Liverpool get the job done, but it wasn't pretty
Liverpool beat Arsenal 3-1 earlier this month to continue their impressive form against their top-six rivals - they are yet to lose to any of them this season.
But as impressive as the Reds have been against those teams around them, they have struggled against sides lower down the table, with all five of their defeats prior to Sunday's game against sides in the bottom half.
Burnley beat Liverpool at Turf Moor back in August and initially had the measure of their opponents in this encounter, although they were aided by a lethargic display by the hosts.
Liverpool did not create a single chance in the opening 30 minutes but their first shot on target resulted in the equaliser and their second produced the winner.
It was ultimately a clinical display by Liverpool but too many players had off days. They needed Philippe Coutinho to be at his creative best to unlock a disciplined Burnley but the midfielder rarely made a telling pass while in attack Divock Origi failed to manage a single shot on goal.
Klopp - and the club's fans - will know the Reds need to become more convincing against sides lower down the table if they are to develop into future title contenders.
More to follow.