Steven Gerrard's second-half brace and further goals from Maxi Rodriguez and Ryan Babel relegated Burnley and kept Liverpool's top-four hopes alive.
Gerrard's first deflected off Burnley defender Leon Cort past Brian Jensen but his second was a fine first-time curling strike from 25-yards out.
Steven Fletcher later hit the post for Burnley with a left-footed half-volley.
Rodriguez fired home from 12-yards after Alberto Aquilani's pass and Ryan Babel beat Jensen when one on one.
The win left Burnley on 27 points, seven points behind West Ham with two matches left to play, while Liverpool remain in seventh but close the gap on fourth-placed Tottenham to two points, though having played a game more.
The victory was Liverpool's first away from Anfield in the league this year and could yet help them seal Champions League football for next season.
With Tottenham losing to Manchester United and Manchester City drawing with Arsenal on Saturday, Liverpool knew they had a great opportunity to close the gap on the teams above them.
Burnley's task was more simple, anything other than a victory would consign them to relegation after just one season in the Premier League.
Liverpool enjoyed the majority of the early possession, but despite a string of Gerrard corners they failed to create one decent chance, with Babel's screwed shot from 20-yards the only real effort of note.
Too often they gave the ball away, more through their own carelessness than Burnley pressure, leading to a disjointed and scrappy game which suited the Clarets.
It took half an hour for the first shot on target and it came from Liverpool's talisman, Gerrard.
Having burst into the box the ball found his path but after a neat swivel and strike from 12-yards, his half half-volley was comfortably saved by Jensen.
Burnley were full of endeavour with Fletcher heading Martin Paterson's right-wing cross from close range over the bar.
More home pressure followed and just before the break an unmarked Jack Cork, on-loan from Chelsea, should have done better when heading Tyrone Mears' right-wing cross straight at Pepe Reina who made a low save.
At the other end, Babel side footed tamely at Jensen when he found himself unmarked at the back post having broken the offside trap.
Liverpool were forced into an early change just after the break with striker Dirk Kuyt, who had been a solitary figure up front, was replaced by Yossi Benayoun.
Soon after, Mears' 20-yard free-kick was well saved by Reina as Burnley's attempts to break the deadlock gained momentum.
But it was Liverpool who got the first goal of the game and, unsurprisingly, it came from Gerrard.
Liverpool's captain played a neat one-two with Benayoun and having cut inside his man he unleashed a right-footed strike from 20-yards which took a huge deflection off Cort before flying past the wrong-footed Jensen and into the corner of the net.
Moments later Gerrard doubled Liverpool's lead with a sublime strike as Burnley's task became that much harder.
Gerrard latched on to Aquilani's knockdown and unleashed a fierce first-time right-footed curling shot from 25-yards which flew into the left-hand corner of Jensen's goal.
Soon after, Nugent tested Reina but his low shot was too tame to beat the Liverpool keeper.
Fletcher's neat turn and shot from the edge of the area was well saved by Reina while at the other end Rodriguez forced a decent stop from Jensen at the foot of his near post.
Fletcher was back in the action again and after controlling a long ball on his chest he was unlucky to see his 12-yard half-volley fly past Reina but bounce back off the post.
Burnley continued to press but Liverpool were that much more clinical and Aquilani's pass superbly opened up the Burnley defence for Maxi to fire right-footed across Jensen into the far corner of the net from 12-yards for 3-0.
And late on there was time for one more with substitute Lucas playing in Babel who beat the offside trap and found himself one on one with Jensen and the Dutchman made no mistake in firing past the Burnley keeper to seal the victory and Burnley's relegation.
Gerrard's first deflected off Burnley defender Leon Cort past Brian Jensen but his second was a fine first-time curling strike from 25-yards out.
Steven Fletcher later hit the post for Burnley with a left-footed half-volley.
Rodriguez fired home from 12-yards after Alberto Aquilani's pass and Ryan Babel beat Jensen when one on one.
The win left Burnley on 27 points, seven points behind West Ham with two matches left to play, while Liverpool remain in seventh but close the gap on fourth-placed Tottenham to two points, though having played a game more.
The victory was Liverpool's first away from Anfield in the league this year and could yet help them seal Champions League football for next season.
With Tottenham losing to Manchester United and Manchester City drawing with Arsenal on Saturday, Liverpool knew they had a great opportunity to close the gap on the teams above them.
Burnley's task was more simple, anything other than a victory would consign them to relegation after just one season in the Premier League.
Liverpool enjoyed the majority of the early possession, but despite a string of Gerrard corners they failed to create one decent chance, with Babel's screwed shot from 20-yards the only real effort of note.
Too often they gave the ball away, more through their own carelessness than Burnley pressure, leading to a disjointed and scrappy game which suited the Clarets.
It took half an hour for the first shot on target and it came from Liverpool's talisman, Gerrard.
Having burst into the box the ball found his path but after a neat swivel and strike from 12-yards, his half half-volley was comfortably saved by Jensen.
Burnley were full of endeavour with Fletcher heading Martin Paterson's right-wing cross from close range over the bar.
More home pressure followed and just before the break an unmarked Jack Cork, on-loan from Chelsea, should have done better when heading Tyrone Mears' right-wing cross straight at Pepe Reina who made a low save.
At the other end, Babel side footed tamely at Jensen when he found himself unmarked at the back post having broken the offside trap.
Liverpool were forced into an early change just after the break with striker Dirk Kuyt, who had been a solitary figure up front, was replaced by Yossi Benayoun.
Soon after, Mears' 20-yard free-kick was well saved by Reina as Burnley's attempts to break the deadlock gained momentum.
But it was Liverpool who got the first goal of the game and, unsurprisingly, it came from Gerrard.
Liverpool's captain played a neat one-two with Benayoun and having cut inside his man he unleashed a right-footed strike from 20-yards which took a huge deflection off Cort before flying past the wrong-footed Jensen and into the corner of the net.
Moments later Gerrard doubled Liverpool's lead with a sublime strike as Burnley's task became that much harder.
Gerrard latched on to Aquilani's knockdown and unleashed a fierce first-time right-footed curling shot from 25-yards which flew into the left-hand corner of Jensen's goal.
Soon after, Nugent tested Reina but his low shot was too tame to beat the Liverpool keeper.
Fletcher's neat turn and shot from the edge of the area was well saved by Reina while at the other end Rodriguez forced a decent stop from Jensen at the foot of his near post.
Fletcher was back in the action again and after controlling a long ball on his chest he was unlucky to see his 12-yard half-volley fly past Reina but bounce back off the post.
Burnley continued to press but Liverpool were that much more clinical and Aquilani's pass superbly opened up the Burnley defence for Maxi to fire right-footed across Jensen into the far corner of the net from 12-yards for 3-0.
And late on there was time for one more with substitute Lucas playing in Babel who beat the offside trap and found himself one on one with Jensen and the Dutchman made no mistake in firing past the Burnley keeper to seal the victory and Burnley's relegation.
