The Reds took the lead in fortuitous circumstances on 33 minutes. Trent Alexander-Arnold's looping cross to the back post took a deflection off Chris Wood and beat goalkeeper Nick Pope.
Sadio Mane doubled the visitors' lead four minutes later. The 27-year-old was played in by Roberto Firmino and placed a low shot past Pope and into the far corner.
Liverpool added a third with 10 minutes of normal time remaining. Firmino thumped a low shot home to become the first Brazilian to score 50 Premier League goals:
Saturday's victory also sees Liverpool set a new record for consecutive Premier League wins:
Liverpool and Burnley both came flying out of the blocks in an entertaining start to the match at Turf Moor.
Fabinho hammered a shot over the crossbar in the first minute before Liverpool goalkeeper Adrian pushed an effort from Wood around the post.
Liverpool then went close again on five minutes. Mane played in Mohamed Salah inside the penalty area, but Pope just managed to tip his curling shot on to the post.
Salah continued to cause problems and was a whisker away from scoring midway through the first half.
The Egypt international was sent through on goal but took a poor first touch. Pope came out to close down the forward, and the ball hit Salah and flew just wide.
The pressure finally told when Alexander-Arnold netted his first goal of the season just after the half-hour mark. The full-back sent in a cross from deep that just took a nick off Wood and beat his own goalkeeper:
Liverpool's second goal needed no such luck. Firmino pounced on a poor pass from Ben Mee and slipped in the overlapping Mane who finished clinically:
The two goals just before half-time put the Reds in total control of the match and allowed the Reds to cruise through the second half.
Salah hit the side-netting from a tight angle just before the hour mark and then shot tamely at Pope when he ought to have passed to Fabinho.
The visitors wrapped up the win with a fine goal that was started and finished by Firmino. The Brazilian swept the ball out to Salah on the right and then raced forwards and fired a precise shot past Pope from the edge of the box.
Liverpool could have had more goals in the closing stages. Salah was denied a goal by a strong tackle from Erik Pieters, and his failure to pass to Mane seemed to anger the 27-year-old.
Mane was substituted shortly afterwards and did not hide his frustration as he left the pitch and was replaced by Divock Origi.
Football writer Joel Rabinowitz offered his view:
It was another comfortable win for Liverpool, who maintain their superb form and return to the top of the Premier League table ahead of champions Manchester City by two points.
Both teams return to Premier League action after the international break on September 14. Liverpool host Newcastle United at Anfield, while Burnley are away at Brighton & Hove Albion.