Ashley Barnes and Phil Bardsley both had chances for Burnley in a goalless first half, while efforts from Naby Keita and James Milner were blocked for Liverpool.
Jack Cork put Burnley ahead with a tap-in after Liverpool failed to clear a corner nine minutes after half-time.
Milner equalised eight minutes later with a low strike from outside the area after good work by Divock Origi.
Roberto Firmino, a substitute along with Mohamed Salah, put Liverpool in front on 69 minutes from Virgil van Dijk's pass, before Xherdan Shaqiri added a third from Salah's pass in stoppage time.
Liverpool are in second on 39 points, while Burnley remain 19th on nine.
Burnley: 8 Dec v Brighton (A)
Liverpool: 8 Dec v AFC Bournemouth (A)