Jurgen Klopp's men were on course for an emphatic rout when Darwin Nunez drew first blood in the sixth minute, but Vincent Kompany's side grew into the game as the second half progressed, only to see their hopes of an unfancied result dashed by a late Diogo Jota insurance goal.
Burnley boss Kompany unsurprisingly kept changes to a minimum following their beating of Fulham - Mike Tresor was the only new inclusion in place of Jacob Bruun Larsen - while Nunez, Harvey Elliott, Ryan Gravenberch and Jarell Quansah were recalled to a rejigged Liverpool XI.
Even with a handful of changes, a galvanised Reds unit bombarded the Burnley backline early doors, as Elliott forced James Trafford into the first stop of the day after just four minutes.
There was nothing that the Clarets goalkeeper could do about a fine Nunez strike in the sixth, though, as Cody Gakpo laid off his attacking partner inside the D, and the Uruguayan curled a sublime strike into the bottom corner for an early deadlock-breaker.
There was no respite for Kompany's beleaguered troops, as Trafford made a stunning fingertip save to deny Mohamed Salah in the ninth minute, before Gakpo tried his luck from close range 12 minutes later and was also thwarted by the Clarets stopper.
The Dutchman did make the Burnley net bulge in the 29th minute, but Paul Tierney's whistle immediately blew for a foul on Charlie Taylor by Nunez, who wagged his finger in disapproval to no avail.
Not content to declare at one in the first half, Liverpool clipped the bar through the lively Salah in the 34th minute, although the Egyptian's effort may not have stood had it gone in due to an apparent handball.
Trafford's save counter stood at an astonishing seven as the two teams headed down the tunnel, but the 21-year-old was left rooted to the spot in the 55th minute, as Elliott played a neat one-two with Gravenberch before opening up his body and guiding a slick side-footed finish into the back of the net from 10 yards.
However, Salah was in an offside position as the ball left Elliott's foot, and following a look at the monitor, Tierney determined that the Egyptian was impeding Trafford's view, and Liverpool had a goal ruled out for the second time.
Salah appeared to have been pushed offside by a claret shirt, but Tierney either took no notice or was uninterested in the incident, and another disallowed effort for Liverpool breathed new life into a hitherto timid Burnley.
Turf Moor found their voice as Liverpool's lead came under serious threat, and the Clarets should have levelled matters in the 68th minute when Wilson Odobert crossed to the back stick, where Johann Berg Gudmundsson was waiting, but the Icelandic winger headed over the top with the goal at his mercy.
Gudmundsson should have been made to pay for that miss by substitute Dominik Szoboszlai when the Hungarian met Curtis Jones's ball to the back post in the 74th minute, but in trying to keep the ball in play with a high foot, the Hungarian inadvertently cleared the ball away from the Burnley goal as another gilt-edged chance went begging.
Odobert was the next to try his luck in the 79th minute, embarking on a sensational solo run before prodding wide of Alisson Becker's far post, three moments before a long-range Szoboszlai effort fizzed over the top.
The frenetic action did not cease in the dying embers, as Trent Alexander-Arnold's venomous 20-yard volley flew a few inches wide in the 86th minute, and a strike from similar distance by Bruun Larsen also bounced to safety a few seconds later after Wataru Endo had carelessly given the ball away.
Liverpool memorably made a mess of a five-on-one counter-attack against Arsenal at the weekend, but a five-on-two situation in the 90th minute was not squandered in similar circumstances, as the Reds belatedly put themselves out of sight.
Making up for his previous error, Endo beat Dara O'Shea to the ball inside the centre circle and managed to pick out the returning Jota, who combined with Luis Diaz down the left before managing to squeeze the ball through Trafford's legs from an incredibly tight angle.
Unless Aston Villa can destroy Manchester United by a seven-goal margin later on Boxing Day, Liverpool will remain at the summit for at least a couple of days until Arsenal return to action on Thursday, while the Clarets remain 19th, five points adrift of safety.
Burnley now close a topsy-turvy 2023 away to Aston Villa on Saturday, while Liverpool are not back in action until New Year's Day, when Newcastle United make the trek to Anfield.