Liverpool headed to Turf Moor without Mohamed Salah and Philippe Coutinho, but Sadio Mane struck a second-half goal to put Jurgen Klopp's side on course for victory.
Sean Dyche's hosts disappointed against a weakened Liverpool side, but Klopp's defence was almost his downfall once again as Johann Berg Gudmundsson struck with two minutes on the clock to set up a grandstand finish.
And there was time for one more goal as Ragnar Klavan finished from close range in injury time to pick up the points.
With top scorer Mohamed Salah and playmaker Philppe Coutinho injured and Roberto Firmino on the bench, the seven changes Klopp made left his side short of ideas, despite Adam Lallana's first league start of the campaign.
Salah, Coutinho, Firmino and Mane have scored 58 of Liverpool's 77 goals this season - 32 of 48 in the Premier League - so their absence from the starting line-up at the ground which has witnessed only six away goals all season was always likely to be a risky strategy.
The Claret's home record, having scored only seven goals themselves, was never likely to lead to an open game and in wet, windy and cold conditions there was little for fans to celebrate the new year with.
Having weathered some early pressure from the hosts in an otherwise ordinary first half, Mane's brilliant goal - only his second in the Premier League since August - lit up the match.
Alex Oxlade-Chamberlain's long-range shot which forced Nick Pope to parry away and Dominic Solanke's heavy touch in the penalty area were the closest Liverpool came to decent chances with most of those falling to the opposition.
Lallana denied Gudmundsson a clear run on goal as Burnley countered from a Liverpool corner, with Scott Arfield seeing one weak header saved and a low, left-footed shot fly inches wide of the far post.
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Early in the second half Mane wanted a penalty for Phil Bardsley's challenge, but referee Roger East waved play on.
On the hour Can picked out Trent Alexander-Arnold wide on the right and Mane controlled his deflected cross with one touch before smashing a left-footed effort into the roof of the net with his second.
Pope then tipped away an Alexander-Arnold drive as the visitors finally found a second gear, but Burnley's threat was not diminished with Ben Mee heading over a corner and Arfield flashing another shot narrowly wide.
The improving Oxlade-Chamberlain had a near-post effort batted behind by Pope, with Simon Mignolet saving substitute Sam Vokes' low header as the game remained more open than Klopp would have liked.
And so it proved as Vokes flicked on a cross for Gudmundsson to equalise, only for Klavan to snatch it back as Klopp's unbeaten record was extended to 16 matches - his best run since coming to England.
Additional reporting by the Press Association.