Ashley Barnes' 83rd-minute penalty – his 100th senior career club goal – inflicted the first league loss on home soil for Jurgen Klopp's side since Crystal Palace in April 2017, after a run of 68 matches undefeated.
The defending champions were again made to pay for their lack of a cutting edge – having left the out-of-form Mohamed Salah and Roberto Firmino on the bench for an hour – and it is now seven hours and 18 minutes since they last found the net in the Premier League.
It was a famous victory for Burnley, whose last win on this ground came in September 1974 courtesy of Ian Brennan's goal.
Last season they were the only side to take a point off Liverpool at home as they raced to the title, and they finished this game with the same amount of possession – 29 per cent – but crucially went one better.
They employed similar tactics of defending resolutely, competing physically and taking the opportunity when it arose.
The same could not be said for Klopp's side, who had 27 attempts but only six on target, and they are now six points behind leaders Manchester United.
A fifth win of the season lifted Burnley to 19 points, but the confidence of the result will be just as important.
With captain Jordan Henderson out with a minor groin problem and Salah and Firmino rested, the onus was put on replacements Divock Origi, with one league goal since December 2019, and Alex Oxlade-Chamberlain.
Neither lived up to the billing and did not last the hour, although not before Origi missed the chance of the night just before half-time.
The Belgium striker's only goal this season came in the 7-2 Carabao Cup victory at Lincoln and his stock falls lower by the week having used up all his credit from that famous Champions League semi-final double against Barcelona in 2019.
An uncharacteristic mis-kick from Clarets captain Ben Mee put Origi clean through, but his lack of game time and confidence showed as he blasted a shot past Nick Pope but against the crossbar when what was required was placement and a calmness under pressure.
It was a chance he just had to take to prove his value as back-up to the established front three.
Pope arrived at Anfield with five clean sheets in his last 10 league games, which did not bode well for Klopp's goal-shy side, but he was not properly tested in the opening 45 minutes.
Sadio Mane had an early header blocked by Dwight McNeil, but there followed a succession of shots from Xherdan Shaqiri, Oxlade-Chamberlain, Origi and Andy Robertson which the England international dealt with comfortably.
Frustrations boiled over at the half-time whistle as Barnes reacted angrily to Fabinho appearing to flick out at him, provoking a small melee, but referee Mike Dean showed the Brazilian a yellow card and VAR agreed.
Meanwhile Klopp and Sean Dyche, who were both halfway down the tunnel, had their own heated exchange
Burnley's one real opportunity came when Alisson dropped a cross under pressure from Chris Wood, but he reacted well to block Barnes' shot.
Just over 10 minutes into the second half and with the game drifting, Klopp sent on Salah and Firmino, with the former straight into the action by forcing Pope into his first decent save low at his near post.
Burnley substitute Johann Berg Gudmundsson blazed wide from close range, but it was not as bad as the Firmino effort which was so off-target compatriot Alisson, 80 yards away, dropped to his knees, held his head and let out a yell.
There were more more shouts of frustration when Alexander-Arnold's cross hit Erik Pieters' arm in the penalty area.
Having spent most of the preceding 83 minutes on the back foot, Burnley were presented with a chance of producing a shock at Anfield and they seized it.
Alisson brought down Barnes and the Clarets forward stepped up to score.
And there was to be no lucky escape for the hosts as Firmino's close-range shot was deflected wide by Mee, who knew little about his intervention.