The Clarets will go in to the international break with four points from as many Premier League fixtures.
It's been an unenviable start for Burnley this term, with trips to the Emirates and Molineux followed by the visit of Liverpool.
The hosts were always going to be up against it when welcoming the only side in English football's top four tiers to boast a 100% record.
The Reds, who lost out on the title by a solitary point to champions Manchester City last season, headed in to the game on the back of 12 successive victories in the Premier League.
The Clarets were more than a match for the European champions in the first half-an-hour at Turf Moor, but trailed at the break as the visitors scored twice in the space of three minutes.
Burnley had battled well against the Premier League leaders, limiting their presence in front of goal to very little, but a freak opener in the 34th minute disrupted the home side's flow.
Sean Dyche's men had taken the lead in five of the last six meetings between the two teams in the top flight and they weren't too far away from stretching that sequence.
Chris Wood timed his run well, beating the offside trap to collect Matt Lowton's ball over the top and, after stepping inside Virgil van Dijk, the striker forced Adrian in to a fine one-handed save.
The hosts would need everything to work in their favour to keep Liverpool out. A combination of Nick Pope's fingertip and the woodwork denied Mo Salah, who had taken Sadio Mane's pass first time.
Pope, who was recalled to Gareth Southgate's England squad earlier in the week, was then off his line in an instant to save at the feet of Salah once the Egyptian international had cushioned Jordan Henderson's pass on the edge of the six yard box.
'Thou shalt not pass' was the message from Burnley's back line; it seemed as though it was going to take something spectacular or out of the ordinary to break that resolve.
Unfortunately for the Clarets, it was the latter. There was nothing wrong with Pope's starting position when Trent Alexander-Arnold shaped to whip in the cross.
However, a wicked deflection off the back of Wood altered the trajectory of the ball, which looped over the head of Pope to give the Reds the lead.
With their spirits wounded the home side imploded, gifting Liverpool their second within minutes.
A lazy pass from Ben Mee to James Tarkowski was cut out by Roberto Firmino and the Brazilian forward slotted the ball in to Mane who, with breath-taking ease, finished in to the far corner.
Ashley Barnes went for broke at the end of the half, looking to add to his four-goal tally, but both attempts from outside the penalty area failed to threaten Adrian's clean sheet bonus.
Burnley carried little threat in a stale second half as the away side managed the game in a calm and controlled manner.
Firmino wrapped up the points with 10 minutes remaining, swiping the ball out of the reach of Pope from 20 yards out after Salah edged goal side of Erik Pieters to assist.
Substitute Jay Rodriguez went close to a consolation late on when taking on both van Dijk and Adrian in the box, but the Spanish stopper recovered well to stick a hand out and repel the shot.