Benayoun scored his first goal after 27 minutes from the edge of the area, and then saw his shot blocked for Dirk Kuyt to score the second before the break.
Steven Gerrard set up Benayoun to tap home the third before substitute Andriy Voronin played him in for his third and Liverpool's fourth near the end.
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 Brian Jensen.
Liverpool's best early chance came when Emiliano Insua's chip was headed a foot wide by Benayoun, but 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.
Liverpool went further ahead when Fernando 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.
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.
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 Stephen Jordan - and he side-footed the ball past an exposed Jensen.