Darwin Núñez snapped a 12-game scoreless streak by plundering the opener with a sweeping first-time finish from the edge of the area.
The visitors looked to have doubled their lead when Cody Gakpo drilled home, but saw his effort disallowed for a foul by Núñez on Charlie Taylor in the set-piece.
The second goal for Jürgen Klopp’s side came early in the second half when Harvey Elliott finished with a finish before VAR intervened again.
That double reprieve saw the Clarets develop a second wind as Jóhann Berg Guðmundsson headed just wide from Wilson Odobert’s delivery.
Diogo Jota came on late for the Reds, replacing Núñez for the first time since suffering a hamstring injury at Manchester City four weeks ago.
The Portuguese forward marked his comeback by curling a shot past James Trafford at his near post in the first minute of second-half extra time.
Liverpool player ratings
GK: Alisson – 6/10 – You’d be forgiven for forgetting Alisson was playing.
RB: Trent Alexander-Arnold – 7/10 – Some smart defensive moments but was mainly involved going forwards. Played some nice passes to try and exploit Burnley’s high line.
CB: Jarell Quansah – 7/10 – A solid centre-back performance. Enjoyed his battle with Burnley’s forwards.
CB: Virgil van Dijk – 8/10 – Some delightful balls over the top to try and make things happen. Had to lead by example in the second half when Burnley started threatening.
LB: Joe Gomez – 7/10 – Gomez is clearly on a quest to score his first senior goal. Pretty active going forwards but had to battle his own attacking shortcomings in the final third.
DM: Wataru Endo – 7/10 – Hardly tested but snuffed out any pressure whenever it came his way. Another really composed performance.
CM: Ryan Gravenberch – 6/10 – Fine. Just fine. Gravenberch was a little sloppy in possession but got away with it because of the level of competition against him.
CM: Harvey Elliott – 7/10 – A classy showing from Elliott, who will have ramped up the pressure on Dominik Szoboszlai in the battle for this midfield spot.
RW: Mohamed Salah – 7/10 – Pressed well and was a constant menace in the final third. Just couldn’t find the luck he needed to make it count.
ST: Darwin Nunez – 8/10 – Finally ended his dry spell with a delightful strike from outside the box.
LW: Cody Gakpo – 8/10 – Really enjoyed being back out on the left wing. Charged at Vitinho at every given moment and was unfortunate not to get his name on the scoresheet as well.
SUB: Curtis Jones (67′ for Gravenberch) – 5/10
SUB: Dominik Szoboszlai (67′ for Elliott) – 5/10
SUB: Luis Diaz (67′ for Gakpo) – 5/10
SUB: Diogo Jota (84′ for Nunez) – 7/10