The Reds made it four in a row at the start of the new season to maintain their status as the only club still with a 100% record, moving back above Manchester City at the top of the fledgling Premier League table.
Jurgen Klopp's side needed a stroke of fortune to get on their way in a tricky-looking fixture as Trent Alexander-Arnold's cross deflected in off Chris Wood in freakish circumstances for the opener, but Sadio Mane quickly added a second to give the visitors full control of the contest.
Roberto Firmino then made sure of the points with his 50th Premier League goal 10 minutes from time, sealing a record-breaking win which sees Klopp achieve something that not even the likes of Bob Paisley, Bill Shankly and Kenny Dalglish managed during their trophy-laden reigns.
The match got off to a spritely start at both ends with Liverpool immediately getting on the front foot as Fabinho fired a long-range effort well over before Burnley hit back with the first clear chance of the contest after less than two minutes.
One long ball forward sent Wood through on goal, and the New Zealander cut back onto his left foot before forcing a smart low save from Adrian in the Liverpool goal.
The visitors came closer still just three minutes later when the front three were all involved as Firmino and Mane combined to find Mohamed Salah, whose instinctive first-time strike crashed against the woodwork via the fingertips of Nick Pope.
The chances did soon dry up as Burnley made life difficult for Liverpool, although Salah came close again midway through the half when he latched on to Jordan Henderson's clever pass only to see the ball ricochet off his shins and narrowly wide after Pope had quickly come off his line to smother the chance.
Klopp's side spent most of the opening half an hour toiling for a way through, though, and it took a bizarre stroke of luck for them to finally find an opener as Alexander-Arnold's deep cross flicked off the back of Wood before looping over Pope and into the far corner, eventually going down as a Wood own goal.
It was a gutting way for Burnley to concede after so much hard work defensively before that, and one quickly became two as the European champions doubled their advantage four minutes later.
This time it was a mistake which proved costly as Ben Mee gifted the ball straight to Firmino, who carried it forward before laying it into the path of the overlapping Mane to finish - his fourth goal in the last four games.
As is to be expected from a Sean Dyche side, Burnley refused to give in and pushed for a way back into the match towards the end of the first half, with the in-form Ashley Barnes missing the target with two shooting opportunities in quick succession.
Liverpool largely managed their lead well, though, and perhaps should have added to it shortly after the hour mark when Salah worked space on the edge of the box but then opted for a tame finish straight at the keeper with his right foot instead of playing through an unmarked Firmino for what would have been a clear chance.
Firmino did have a sight of goal shortly afterwards, failing to catch an acrobatic scissor kick perfectly, before the newly-crowned UEFA Player of the Year Virgil van Dijk attempted an even more ambitious effort with a 40-yard half-volley which flew comfortably over.
Burnley knew that one goal against a defence that had not kept a clean sheet so far this season would give them a sniff, but chances were at a premium and James Tarkowski sent one of their best over the top with a header 17 minutes from time.
The hosts had little option but to throw caution to the wind in their search for a route back into the game, and that left more space at the other end with Alex Oxlade-Chamberlain firing a powerful effort into the arms of Pope before Liverpool added their third.
While it was a goal which secured a notable piece of club history, Firmino claimed an individual place in the history books too as he became the first Brazilian to reach a half-century of Premier League goals, drilling a first-time finish into the bottom corner after Salah had left the ball to him on the edge of the area.
Salah's desire to join his attacking teammates on the scoresheet was clear to see, although once again he was perhaps too selfish shortly afterwards when he turned down a pass to the unmarked Mane in favour of trying to engineer his own shot, allowing Erik Pieters to get back and make the challenge.
Mane appeared visibly upset with his teammate after being substituted moments later, and Klopp might have shared his anger had Liverpool's first clean sheet of the campaign been surrendered with only one minute left of normal time.
Jay Rodriguez managed to skip away from Van Dijk and was left with only Adrian to beat, but the Spaniard stayed big even after committing too early to thwart the Burnley sub.
There was still time for Liverpool to threaten a fourth as Salah stabbed a pass through for Georginio Wijnaldum, who tried to take the ball around Pope only for the Burnley keeper to get an important hand to it.
The visitors were more than content to settle for a three-goal win, though - as well as that first clean sheet of the season - as they continued their fine form to ensure that they will go into the first international break top of the table and still perfect.
As well as a record 13th successive win, Liverpool also become only the second team in English top-flight history to win 13 in a row while scoring more than once in each game, joining Tottenham Hotspur from 1960 in achieving that feat.
BURNLEY (4-4-2): Pope; Lowton, Tarkowski, Mee, Pieters; Lennon, Westwood, Cork, McNeil; Barnes (Rodriguez 73'), Wood
LIVERPOOL (4-3-3): Adrian; Alexander-Arnold, Matip, Van Dijk, Robertson; Henderson (Oxlade-Chamberlain 71'), Fabinho, Wijnaldum; Salah, Firmino (Shaqiri 85'), Mane (Origi 85')