The top-flight new boys had started the season so well with wins over Manchester United and Everton but have now suffered successive defeats at Chelsea and now Liverpool - conceding seven in the process.
They could not live with a vibrant Liverpool, who kept up an impressive high tempo throughout and are slowly wiping away the memory of those two early league defeats to Tottenham and Aston Villa.
Benayoun scored his first from the edge of the box, and saw his shot parried by the overworked Brian Jensen for Dirk Kuyt to add the second before the break.
Steven Gerrard's pass set up Benayoun for his second, a close-range tap-in, before the Israeli was sent clear by substitute Andriy Voronin for his third and Liverpool's fourth.
Rafael Benitez made two changes from the side which won at Bolton a fortnight ago, with Martin Skrtel returning to central defence in place of Greece international Sotirios Kyrgiakos.
With Javier Mascherano suffering a muscle injury while on international duty with Argentina and not arriving back until late Friday, Benayoun came into the side.
And what a tremendous impact he made, with a goal and a display of running power that Burnley never came to terms with.
The Clarets, in their first league visit to Anfield for 33 years, retained the side that lost at Chelsea last time out, with new loan signing David Nugent - a former Liverpool trainee - on the bench.
The absence of Mascherano meant that Steven Gerrard played in a much deeper role, alongside Lucas in the centre of midfield, with Benayoun and Kuyt left to play just behind Fernando Torres.
Burnley started brightly with Martin Paterson firing wide from the edge of the box, while Kuyt produced a low shot at the other end that produced the first of many saves by Jensen.
Liverpool slowly built up possession, moving the ball quickly and Torres saw a curling 20-yard shot clear the far post.
But Burnley had Steven Fletcher and Paterson interchanging positions up front and the visitors were quick and dangerous on the break.
Liverpool's best early chance came when Emiliano Insua's chip was headed a foot wide by Benayoun, the beginning of a purple patch from the 29-year-old.
The lively Israel captain broke the deadlock with a gem of a goal after 27 minutes.
He surged from midfield, turned away from Graham Alexander and moved into the box to fire across Jensen and into the bottom corner.
Benayoun almost added another a minute later when he met Albert Riera's cross with a low header that Jensen blocked, the Burnley goalkeeper also keeping out Kuyt's follow-up.
Jensen had to keep out drives from Benayoun and Lucas before Paterson forced his way past Gerrard to unleash a rising drive that was held by Jose Reina.
But Liverpool went further ahead after 40 minutes with a dynamic five-man break from defence.
Torres and Insua combined to set up Benayoun for a 20-yard shot that Jensen blocked, but only as far as Kuyt who clipped home from five yards.
The increasingly busy Jensen had to produce the save of the half to keep out a Gerrard 20-yard drive before the break.
Liverpool kept up their assault after the interval with Riera seeing two efforts blocked while Jensen saved from a Kuyt flick.
Stephen Jordan was booked for a block on Torres before Burnley sent on Chris Eagles to replace Robbie Blake.
Torres was starting to stretch a tiring Burnley defence by now and only Tyrone Mears' fine saving tackle stopped the Spaniard in full flow.
But after 61 minutes Torres' pass played in a fiercely determined Gerrard, who battered his way past two men in the box to find Benayoun on the far post for a tap-in.
Liverpool sent on Swiss defender Philipp Degen for Glen Johnson and Voronin for the hard-running Kuyt, the game well won by now, and Burnley responded by giving Nugent his debut for Paterson.
Benayoun had an effort ruled out for offside and Gerrard forced his way through again to hit a post, before David Ngog took over from Torres.
There was still time for Benayoun to secure his hat-trick after 81 minutes. Voronin's pass sent him clear in the box - played onside by Jordan - and he side-footed the ball past an exposed Jensen.
Eagles was booked for a foul on Benayoun, Jensen saved again from Gerrard and Voronin missed when clear, with Burnley just wanting it all to end.