With two matches to play the Clarets cannot now catch 17th-placed West Ham and return to the Championship after just one season in the top flight.
Steven Gerrard's first goal in the 51st minute was deflected off Burnley centre-back Leon Cort but his 25-yard strike seven minutes later was inch-perfect and Maxi Rodriguez and Ryan Babel wrapped things up.
The win, Liverpool's first in eight away league matches this year, kept Rafael Benitez's side in the hunt for next season's Champions League but only just.
Tottenham are two points ahead on 64, with a match in hand, and Liverpool must win their remaining two games and hope neither Spurs, Manchester City nor Aston Villa amass more than 68 points - and even then it could come down to goal difference.
Burnley, however, must now face up to a return to the second tier of English football, which has been on the cards ever since the departure of former manager Owen Coyle to Bolton in January.
By that time they were already on the slide but the loss of the inspirational Scot was a major blow and the decision to bring in Brian Laws failed to turn things around.
Whether anyone would have been able to save the Clarets on the shoestring budget and limited squad available is questionable.
But opting to employ Laws just a month after he had left Sheffield Wednesday having taken them to the brink of relegation from the Championship now seems, at best, a mis-judgment.
Laws has lost 14 of his 17 games in charge and the Clarets have won just seven games all season and not kept a clean sheet since Halloween.
At least the frugal nature of the sensible financial planning at Turf Moor means Burnley do not have the money worries of Hull, who are also doomed to relegation because of their dreadful goal difference.
The first half summed up exactly why they were heading back to the Championship as, having weathered the early pressure, they missed two gilt-edged headers which should have put them 2-0 up at the break.
Steven Fletcher out-jumped Jamie Carragher in the 27th-minute but planted Martin Paterson's right-wing cross over while on-loan Chelsea midfielder Jack Cork's seven-yard effort from Tyrone Mears' centre was straight at goalkeeper Jose Reina.
Prior to that the closest Liverpool had come to scoring was when Gerrard's deflected volley hit referee Phil Dowd before the Liverpool captain's shot on the turn from 12 yards was easily saved by goalkeeper Brian Jensen.
The break appeared not to have dampened Burnley's enthusiasm and Fletcher almost capitalised on confusion between Reina, Daniel Ayala - the teenaged centre-back a surprise inclusion - and Daniel Agger when Cork swung in a left-wing cross seconds after the restart.
Liverpool's problems up front worsened when Dirk Kuyt was forced off with a calf injury, meaning Babel was pushed into the lone striker role to allow Yossi Benayoun to come on and play on the left.
The Israeli had a hand in the opening goal but luck played a greater part in the 51st minute.
Benayoun and Alberto Aquilani combined in the centre of the pitch to give Gerrard the ball in the inside-left position.
With Graham Alexander backing away he switched the ball on to his right foot before unleashing a 20-yard shot which deflected off Cort and inside Jensen's right post.
Fortune also contributed to Gerrard's second seven minutes later when Glen Johnson and Rodriguez progressed a move down the right to Aquilani who slipped as he tried to turn inside.
However, the ball ran to Gerrard, whose 25-yard strike owed nothing to luck and everything to talent as he curled a shot perfectly through the gap between Jensen's outstretched fingers and the left post.
Fletcher almost pulled one back in the 72nd minute when his left-footed shot cannoned back off the post, although justice was probably done as the striker appeared to control the ball with his hand.
Two minutes later Burnley's fate was sealed as Aquilani played in Rodriguez on the right of the penalty area and he clipped home his long-awaited first goal since signing from Atletico Madrid in January.
In the third minute of added time Babel sprang the offside trap to race clear and fire in a fourth before Turf Moor rose to acclaim their departing heroes.