Goals at either end of the game from Ashley Westwood and substitute Johann Berg Gudmundsson were not enough to gain the Clarets any reward at Anfield as Liverpool re-ignited their title challenge.
Two goals apiece from Roberto Firmino and Mane Sane hauled Liverpool back to within a point of defending champions and leaders Manchester City as they try to bring the championship back to Merseyside for the first time in 39 years.
But victory was not achieved without a few concerns as Sean Dyche’s men asked some serious questions of the wobbling Reds before a hat-trick of errors ultimately proved too great a burden to overcome as Burnley slipped to a third successive defeat to stay just two points above the relegation places.
The Clarets, without a win at Anfield since 1974, made the perfect start in their quest to end that wait and restore some breathing space at the bottom end of the table.
Westwood ended his own personal drought when he curled the ball directly from a corner under the bar and inside the far post – Liverpool keeper Allison earning a booking for dissent after claiming he had been held down on the line.
That goal just six minutes in raised the anxiety levels around Anfield as Burnley threatened to make it another difficult afternoon for the Reds, whose title surge had been de-railed by four draws in their previous six league games.
But, crucially, the lead lasted less than a quarter of an hour and two goals in 10 minutes restored belief to the Liverpool ranks as they led at half-time.
Frustratingly for Burnley, they will feel both goals could have been avoided as Klopp’s side – unbeaten in 35 home Premier League games – were given a helping hand.
For the first, a combination of Tom Heaton and James Tarkowski was unable to cut out a low cross from Mohamed Salah, leaving Firmino - who had been allowed to grab the ball to claim a free-kick in the build-up - with a tap-in to mark his return to starting duty.
And further hesitancy in attempting to eradicate danger then saw Phil Bardsley forced into a hurried clearance which was charged down by Adam Lallana, with Charlie Taylor’s fine covering tackle on Salah only directing the ball into path of Mane, who scored at Anfield for a sixth successive home league game – the fifth Liverpool player to do so.
The Clarets also started the second half positively, but with Liverpool’s players swarming around the ball whenever Burnley had possession it was tough to create any real openings.
Again, teenager Dwight McNeil – keeping his place in an unchanged starting line-up – stood out on one of the biggest stages in the country as he constantly looked to make something happen with his sniping runs and teasing deliveries.
And while there was just one goal in it, the Clarets remained in touching distance of a what would have been a big result.
If anything, the momentum was with the Clarets but mid-way through the second half disaster struck as Liverpool were gifted a third goal which seemed to take them out of reach.
Heaton’s poor goal-kick went straight to Salah, who sprinted into Burnley’s penalty area where another fine Taylor tackle stopped the Egyptian but only teed up Firminho to slot home his second of the game.
That might have been it, but Dyche made a double change with Gudmundsson and Peter Crouch coming on, the latter given a tremendous reception by the home fans who remembered his contribution in Liverpool colours.
Matej Vydra was also introduced and the three gave the Clarets late impetus and a goal for Gudmundsson, who swept home his third goal of the season from Vydra’s astute pass a miniute into stoppage time.
With a couple of minutes to go, Burnley then pressed for a dramatic late leveller as nerves once again engulfed the stadium.
But with the final kick of the game Liverpool finally made sure as Mane was released on the break and touched the ball around Heaton before slotting home to keep the Reds’ title hopes alive.
Liverpool: Alisson; Alexander-Arnold, Matip, van Dijk, Robertson; Wijnaldum (Henderson 67), Fabinho, Lallana (Keita 76); Salah, Firmino, Mane. Substitutes: Lovren, Sturridge, Mignolet, Shaqiri, Origi.
Booked: Alisson, Matip
Burnley: Heaton; Bardsley, Tarkowski, Mee, Taylor; Hendrick (Gudmundsson 79), Cork, Westwood, McNeil; Barnes (Vydra 86), Wood (Crouch 79). Substitutes: Lowton, Brady, Gibson, Hart.
Referee: Andre Marriner