Liverpool 2 Burnley 1
Burnley scratched a 42-year itch but could not end another long wait as Liverpool came from behind to extend their unbeaten home run against the Clarets.
Ashley Barnes scored in front of the Kop to give Burnley an early lead and a first goal at Anfield in nine attempts stretching back to 1975.
A first win at Liverpool since a year before that and a first double over the Reds since 1930 was then a real possibility as Sean Dyche's side again went blow-for-blow with one of the Premier League heavyweights.
But an equaliser from Giorginio Wijnaldum with the last kick of the first half got Liverpool off the ropes and ultimately proved the turning point as Jurgen Klop's Champions League hopefuls added their name to that of Tottenham, Manchester City and Arsenal to have scraped past Burnley by a single goal on their own pitch.
Wijnaldum's effort was the Reds' first shot on target and they were to manage only one more before the 90th-minute as the Clarets again belied their status as one of only two top-flight sides not to have won away this season.
But that second-half strike - expertly dispatched from long range with an hour gone by Emre Can - was enough to give Liverpool victory and maintain the fixture's status quo.
The Clarets could not have made a better start as they looked to defy the odds and enjoy a first Anfield success since Ian Brennan's goal brought home the points in 1974.
Dyche's men took the game to a Liverpool side which had beaten Tottenham and Arsenal in their last two home games to reclaim a top-four place.
George Boyd flashed a half-volley wide and Barnes just failed to reach a dangerous cross from Andre Gray, whose running in behind the Reds' back four caused them a host of problems.
And the goal Burnley had waited since 1975 to score on this ground came with seven minutes gone as the Clarets took a deserved lead.
The build-up was top-notch as the visitors worked an opening clinically created by Matt Lowton's sublime through-ball which Barnes met perfectly on the stretch with the outside of his right boot to steer home his fifth goal of the season.
Burnley did not just sit on their lead as Barnes and Gray continued to ask questions of the Liverpool back line and the Clarets' midfield, including the re-called Scott Arfield, worked overtime to keep moving the ball forward stop Liverpool getting up a head of steam.
The Clarets' back four superbly protected Tom Heaton as Liverpool were denied a clear goal-scoring chance until they struck with the last kick off the first half to make it 1-1.
Even then it looked as though the danger had been snuffed out when Ben Mee blocked Wijnaldum's initial flick from Divock Origi's cross.
But when the ball dropped kindly back to the Netherlands international, he was able to pick his spot and side-foot home past an unsighted Heaton to partially apease the frustrated home fans.
Despite that set-back, the Clarets started the second half positively as they attacked the end housing a superb away support.
Barnes had a couple of efforts off target as Liverpool continued to struggle to deal with the Clarets' front man, who was back after a one-match ban.
But Liverpool were seeing more of the ball themselves and gradually pushing Burnley deeper.
And their pressure paid off when Can fired Klopp's side in front.
The Germany international picked the ball up 30 yards from goal following a throw-in and strode forward before curling a low shot just beyond Heaton, who saw it late, and inside the keeper's left-hand post.
The Clarets' upbeat response saw a Barnes shot charged down in front of goal after Ben Mee had headed down an Arfield free-kick into the striker's path.
And with 20 minutes to go, Dyche sent on Robbie Brady to try and conjur up a leveler.
Brady had one sight of goal after a free-kick was headed his way but with his right foot, the wideman was unable to locate the target from a testing angle.
Dyche also sent Sam Vokes into the fray and in the last minute gave a senior debut to teenager Dan Agyei as the Clarets tried to turn on maximum firepower against a back-pedaling Liverpool side which was increasingly being forced to try and hang on to their slender lead.
But after Heaton had denied Sadio Mane as the Reds mounted the rare threat of a third goal on the break, the Clarets' final chance fell the way of Lowton in stoppage time with a close-range volley on the turn from the full-back flying just over the bar to continue the Clarets' away-day frustrations.
Burnley: Heaton, Lowton, Keane, Mee, Ward, Boyd (Brady 72), Hendrick, Barton, Arfield (Agyei 90), Gray (Vokes 79), Barnes. Substitutes: Robinson, Darikwa, Tarkowski, Westwood.
Booked: Mee, Barton
Liverpool: Mignolet, Clyne, Klavan, Matip, Milner, Can, Wijnaldum, Lallana, Coutinho (Woodburn 60), Mane, Origi (Lucas 79). Substitutes: Karius, Gomez, Moreno, Wilson, Alexander-Arnold.
Booked: Can, Lallana.
Referee: Craig Pawson