The Clarets continued their remarkable record away from home as they secured a first ever Premier League point on Merseyside.
Burnley had gained nothing at Anfield since 1974 and suffered a similar pointless record across Stanley Park at Everton in their previous three Premier League campaigns.
But after winning at Chelsea and drawing at Tottenham, Sean Dyche’s side can now add Anfield to the list of grounds where they have taken some reward in a superb start to the season - with a trip to Everton next up on the away-day map.
Scott Arfield gave Sean Dyche’s men the lead with his first goal in almost a year to puncture Liverpool's 125th birthday celebrations.
And while that lead lasted only a few minutes – Mohamed Salah making it 1-1 at half-time – the Clarets defended superbly to stop Liverpool breaking through again before almost pinching a famous win themselves through inspirational stand-in skipper Ben Mee in a breathless finish.
The Clarets reverted to the formation that had brought them so much joy on the road so far as Arfield returned to starting duty and Chris Wood got the nod to start in the lone striking role after two goals in his first two appearances.
There was also a full Premier League debut for Nick Pope in goal in place of the injured Tom Heaton, with Pope’s first top-flight start coming against the backdrop of the Kop as Liverpool attacked their favourite end before the break.
It could have been a baptism of fire as the Reds, showing seven changes to the side that drew 2-2 with Sevilla in the Champions League in mid-week, looked to bounce back from last weekend’s 5-0 beating at Manchester City.
With Philippe Coutinho seeing plenty of the ball on his first of the season following a summer of transfer speculation the home side dominated for long periods of the first half.
But Pope was well protected by the white shirts in front of him as the Clarets worked over-time to keep Liverpool at bay.
And the Clarets’ keeper had not had a meaningful shot to save before Burnley grabbed an unlikely lead through Arfield’s classy finish with 26 minutes gone.
Robbie Brady had got through some sterling work down Burnley’s left flank and he was instrumental in opening up the Liverpool defence for Arfield to claim his first goal since a winner against Everton at Turf Moor 11 months ago.
Brady’s success in an aerial duel from James Tarkowksi’s long ball forward allowed Wood to help the ball into the path of Arfield.
And the first-time finish from the Canadian international was top-class as he passed the ball into the bottom right-hand corner of Simon Mignolet’s net from 15 yards.
Crucially, though, Burnley were unable to hold onto their lead from more than three minutes as Liverpool hit back to make it 1-1 through dangerman Salah.
The former Chelsea and Roma winger had been a constant threat and when he managed to get in behind the Clarets’ back line to meet Emre Can’s long ball, a couple of touches and shot on the turn was all Liverpool's record buy needed to bury the equaliser.
The momentum was back with the Merseysiders and stayed that way after the break as Burnley struggled to gain any sustained possession against a wave of red shirts.
Daniel Sturridge forced Pope – who was booked for time-wasting – to beat away a shot from outside the penalty area and Tarkowski made a fine block from James Milner as the pressure increased.
Dyche threw on Ashley Barnes – a scorer here last season to end the Clarets’ 46-year wait for a league goal at Anfield - as an added outlet to try and reverse the flow.
His added presence up front bought some breathing space and hope of a second goal as Liverpool’s back line was asked some questions, Brady testing Mignolet after being released by the tireless Cork.
And with 10 minutes to go, the Clarets twice almost grabbed a dramatic winner as they suddenly looked the ones full of running.
Acting captain Mee was the man denied, both times from Brady corners, with his first header cleared off the line by Joel Matip and his second blocked by the legs of Mignolet at his near post.
But there was still for drama at the other end as Liverpool pushed desperately for maximum points.
An unsighted Pope made an excellent save from teenage full-back Trent Alexander-Arnold to cap a memorable afternoon for the young keeper.
And with the Clarets at full stretch, substitute Dominic Solanke was inches away from his first Premier League goal with a close-range effort he thudded against the bar - Pope then clawing away the follow-up - as Burnley held out magnificently to secure another big point and keep their unbeaten away start intact.
Liverpool: Mignolet, Alexander-Arnold, Matip, Klavan, Robertson, Milner, Can, Coutinho (Solanke 78), Salah, Surridge, Firmino (Oxlade-Chamberlain 78). Substitutes: Karius, Wijnaldum, Lovren, Henderson, Moreno.
Burnley: Pope, Lowton, Tarkowski, Mee, Ward, Gudmundsson (Barnes 60), Cork, Defour (Westwood 87), Brady, Arfield, Wood (Vokes 87). Substitutes: Legzdins, Bardsley, Long, Walters.
Booked: Pope, Cork.
Referee: Roger East