Johann Berg Gudmundsson’s first goal of the season was not enough to earn the Clarets a point as their thrilling comeback was cruelly cut short by an injury-time winner from Ragnar Klavan.
Gudmundsson dived in to make it 1-1 with just a couple of minutes to go as Burnley deservedly hit back from Sadio Mane’s opener in a storming finish.
But deep in stoppage time Klavan bundled the ball home following a free-kick to give the defender a first goal in a Liverpool shirt and a record unbeaten run under manager Jurgen Klopp.
It was rough justice on the Clarets, who again more than matched one of the Premier League heavyweights before being denied a big result right at the death – Liverpool matching the late goal Manchester United scored on Boxing Day to deny Sean Dyche’s men a famous win and leaving them with just two points to show from a major effort over the four-game holiday programme.
The return of James Tarkowski, wearing a light cast on the hand he broke at Brighton a fortnight ago, following his three-match ban saw the Clarets’ only change as they completed their busy festive schedule.
Liverpool, on the other hand, saw fit to make seven changes after extending their unbeaten run to 15 games with a weekend win over Leicester City.
Injured top scorer Mo Salah, who scored both goals against the Foxes, was among those to miss out, with Philippe Coutinho not in the Reds’ squad at all, and Roberto Firmino starting on the bench alongside former Turf Moor favourite Danny Ings.
There was a first league start of the season, though, for Adam Lallana and the England international was a central figure in a goalless first half.
The midfielder had an early sight of goal he failed to take and it was his cross-field pass for Alex Oxlade-Chamberlain which brought Liverpool’s only effort on target before the break, Nick Pope getting down to beat away a skidding drive.
But Lallana’s most critical intervention came at the other end as his superb recovery run and perfectly-timed tackle just stopped Berg Gudmundsson in his tracks as the Clarets’ wideman almost got onto a lovely pass from Ashley Barnes for a clear run at goal.
Barnes caused his usual quota of problems, shooting wide early on after charging down an attempted clearance from Trent Alexander-Arnold.
And as the Clarets pressed for the opening goal with a dominant spell just before the break he set up a chance for Scott Arfield, who was only inches away from giving Burnley the lead.
Arfield – who had scored the Clarets’ goal in a 1-1 draw at Anfield in September – had earlier glanced a header goalwards from Steven Defour’s floated pass which goalkeeper Simon Mignolet grabbed to his right-hand side.
But the Reds’ captain would have had little chance had Arfield’s first-time finish from a Barnes’ knock-down gone just the other side of the post as it skimmed across a rain-sodden surface in horrible wet and windy conditions.
The Clarets continued to probe at the start of the second half but couldn’t force an opening and with an hour gone Liverpool grabbed the lead with a quality goal on the break.
Oxlade-Chamberlain launched the counter-attack by spreading the ball wide to Alexander-Arnold and even though the teenager’s low cross was behind Mane, his touch and shot on the turn gave Pope no chance as the Senegal international crashed home his eighth goal of the season from the edge of the penalty area.
Burnley’s response was positive with Defour testing Mignolet with a low strike and with 20 minutes to Dyche sent on Sam Vokes – whose first Premier League goal had set up the Clarets’ first league win over Liverpool in 42 years last season – in place of Jeff Hendrick.
The visitors came close to killing the game through Alexander-Arnold as his long-range effort was top-corner bound until Pope made a fine save to palm the ball away.
But Vokes was then involved as the Clarets almost grabbed an equaliser, keeping the ball alive following a free-kick awarded when Joe Gomez wrestled Gudmundsson to the ground for Barnes to send a left-footed volley just past the far post.
With the Clarets committed to attack, it took another fine save from Pope to keep out an Oxlade-Chamberlain effort as Liverpool again threatened on the break.
But the Clarets refused to give up in a barnstorming finish and, backed by their biggest crowd of the season, got the goal they deserved to make it 1-1.
Vokes had just been denied by Mignolet with an excellent stop to keep out a downward header from Gudmundsson’s cross when the big striker returned the favour for Gudmundsson to deservedly level with two minutes to go.
Charlie Taylor did well to swing over a cross which Vokes met with an angled header and Gudmundsson arrived perfectly at the far post to nod home.
But ecstasy turned to agony in the fourth minute of stoppage time when Liverpool pinched a winner.
Emre Can’s free-kick was nodded on by Dejan Lovren and Klavan dived to head in from virtually on the line to cruelly deny the Clarets a point to start the New Year.
Burnley: Pope, Bardsley, Tarkowski, Mee, Taylor, Gudmundsson, Cork, Defour, Arfield (Wells 86), Hendrick (Vokes 70), Barnes. Substitutes: Lindegaard, Lowton, Westwood, Walters, Long.
Liverpool: Mignolet, Alexander-Arnold, Lovren, Klavan, Gomez, Wijnaldum, Can, Mane (Firmino 70), Lallana (Milner 86), Oxlade-Chamberlain (Matip 90), Solanke. Substitutes: Karius, Robertson, Ings, Woodburn.
Referee: Roger East