Darwin Nunez scored his eighth goal of the season for Liverpool
Jurgen Klopp's side have only been beaten once so far this season and went two points clear of Arsenal at the summit despite some wastefulness in front of goal.
The Reds got the perfect start by netting after just six minutes as Darwin Nunez guided in a lovely finish from the edge of the box to end a run of 12 club games without a goal.
The impressive James Trafford was kept busy in the Burnley goal and had already smothered Harvey Elliott's low drive before making a stunning fingertip stop to deny Mohamed Salah.
The wave of attacks from the Reds was unrelenting and Cody Gakpo had a goal disallowed for Nunez's foul on Charlie Taylor, while Salah rattled the crossbar - all before half-time.
Elliott also had an effort chalked off for Salah interfering with play in a offside position, but the visitors did finally get their second when Diogo Jota smacked home in the final minute.
Burnley, meanwhile, failed to work Alisson into a single save and are in deep trouble in the bottom three, now five points adrift of safety.
Aston Villa will go level on points with Liverpool if they beat Manchester United later on Tuesday (20:00 GMT kick-off), while Arsenal can regain top spot with success over West Ham on Thursday.
-- Courtesy of BBC Sport