After a frustrating opening half-hour, Liverpool took the lead with some help from the visitors as Diogo Jota headed home from a Trent Alexander-Arnold corner that goalkeeper James Trafford misjudged.
But on the stroke of half-time, the Clarets restored parity through Dar O’Shea, who powerfully headed Josh Brownhill’s corner past his compatriot Caoimhín Kelleher.
Burnley’s joy was short-lived, however, as they quickly fell behind again after the restart when Elliott, who had replaced the unfit Trent Alexander-Arnold at half-time, made an immediate impact by crossing for Luis Díaz to finish.
Both Trafford and Kelleher then produced fantastic saves to keep the score at 2-1 before David Fofana, fresh from last week’s brace, fluffed his lines when he got a big mistake on goal and was booked for it with 10 minutes to go.
Darwin Núñez, who has attempted more shots than any other Premier League player this season and has had a poor run in front of goal of late, put the Reds two ahead by heading home another Elliott cross to put his side two points ahead. Manchester City, who have the game in their hands.
Liverpool player ratings
GK: Caoimhin Kelleher – 7/10 – Some great saves to deny Burnley in Alisson’s absence.
RB: Trent Alexander-Arnold – 7/10 – Set up Jota’s opener with a delightful corner and frequently showed his range of passing from deep and advanced positions. Substituted at the break.
CB: Jarell Quansah – 6/10 – Booked for a late challenge on Odobert in the first half and not his most comfortable display in a Liverpool shirt.
CB: Virgil van Dijk – 7/10 – Part of a defence that was a little disjointed, conceding more opportunities than they would have liked. However, he put in a largely solid display.
LB: Andy Robertson – 7/10 – Given how little football he has played since injuring himself in October, it was a good performance from the Scotsman. Offered threat down the left flank going forward.
CM: Alexis Mac Allister – 7/10 – A solid showing from the Argentine, who helped Liverpool slow the game down when needed.
CM: Wataru Endo – 7/10 – A little off the pace in the first half in his first start since his return from the Asian Cup, but grew into the game. A key creator in the second period with some neat forward passing.
CM: Curtis Jones – 6/10 – Not his brightest display in either half but a decent overall outing.
RW: Luis Diaz – 8/10 – Lively throughout and was rewarded for a spirited effort with a goal. A constant menace in the final third.
ST: Darwin Nunez – 7/10 – Full of endeavour, as he always is, but sometimes lacked quality in the Burnley half. However, his header was a lovely finish.
LW: Diogo Jota – 7/10 – The Portuguese just has a knack for scoring with his head, this time rising highest from a corner to give Liverpool the lead. Could have perhaps had more on another day.
SUB: Harvey Elliott (46′ for Alexander-Arnold) – 8/10 – Provided two assists with delicious deliveries into the penalty area. A great impact off the bench.
SUB: Cody Gakpo (82′ for Diaz) – N/A
SUB: James McConnell (90′ for Endo) – N/A
SUB: Kostas Tsimikas (90′ for Robertson) – N/A
SUB: Bobby Clark (90′ for Mac Allister) – N/A