It's the Reds' first loss at Anfield in the league since April 2017 – ending the club-record unbeaten streak at 68 – as Ashley Barnes fired in an 83rd-minute penalty.
Divock Origi came closest to delivering a breakthrough for the hosts but struck the crossbar towards the end of the first half.
Despite dominating possession and forcing goalkeeper Nick Pope into a number of saves, Jürgen Klopp's men were unable to take anything from the game.
Team news
Joel Matip made his comeback in the fixture as he replaced the injured Jordan Henderson at centre-half.
Meanwhile, Alex Oxlade-Chamberlain and Origi were handed starts, coming in for Mohamed Salah and Roberto Firmino, who both began on the bench.
First half
The Reds made a strong start to proceedings and could've gone ahead after seven minutes. However, Sadio Mane's goal-bound header off a corner was blocked by Dwight McNeil.
At the other end, Burnley created their first opportunity, which required Alisson Becker to make a smart intervention. The No.1 initially fumbled a cross aimed at Chris Wood but pounced on the rebound to deny Barnes' close-range shot.
Liverpool dominated the first half and Xherdan Shaqiri went close to providing an opener with a powerful effort from distance going fractionally wide.
The hosts' next two strikes at goal forced Pope into saves. Oxlade-Chamberlain's drilled shot went straight at the Burnley stopper before he got down low to keep out a hit from Origi.
Origi tested Pope again just past the half-hour mark with a shot right down the middle of the goal following a Mane-led counter-attack.
In minute 39, Andy Robertson was played in down the left and finished off the spell of pressure with another strike kept out by the gloves of Pope.
But the best chance of the half arrived two minutes before half-time when Origi pounced on a mistake from Ben Mee and ran through on goal. The striker tried to side-foot his finish in the top right corner but it struck the crossbar and bounced clear.
Second half
It was more of the same following the interval, with Pope keeping out a well-struck effort from Trent Alexander-Arnold that was heading in.
Salah and Firmino were introduced to the contest in the 57th minute, and the former came close to making an instant impact.
Georginio Wijnaldum picked up possession off Alisson and carried the ball all the way to the Burnley penalty area. The No.5 laid the ball off to Salah, who was then denied at the near post by Pope.
After Burnley went close through Johann Gudmundsson, Firmino didn't connect properly when presented with a chance in front of goal and flashed wide following good pressure from Robertson.
The deadlock was finally broken late on after Alisson was adjudged to have brought down Barnes in the penalty area. The Burnley forward took responsibility from the spot himself and converted.
As Liverpool searched for a way back into the affair, Firmino attempted to flick an Alexander-Arnold shot towards goal but couldn't angle it in.