The Portuguese forward opened the scoring in the 18th minute when stealing a march on Ben Mee to head a Konstantinos Tsmikas cross past Nick Pope.
The Clarets had chances either side of the deadlock being broken as Alisson beat away efforts from Chris Wood and Dwight McNeil.
Mo Salah thought he'd doubled the lead when slotting home from Harvey Elliott's slide rule pass in front of The Kop, but the finish was chalked off for offside following a VAR check.
Ashley Barnes then had the ball in the back of the net for Burnley after the break, converting Matt Lowton's delivery, only for the linesman's flag to intervene once more.
Dwight McNeil stuck out a leg to prevent another Salah effort from crossing the line and Mane's effort shortly after was dealt with unconventionally by Pope.
However, the home side doubled their lead in the 69th minute when Mane hit Trent Alexander-Arnold's flick first time.
Sean Dyche's side had opportunities to reduce the arrears late on only for Virgil van Dijk to block Jay Rodriguez's strike before Alisson saved at the feet of Ashley Barnes.
9. Josh Brownhill 6.5
Renowned for his ability to intercept possession and he did it with aplomb at Anfield, breaking up play on the edge of both boxes. The midfielder's excellence in breaking up play was matched by his energy and quality from set-pieces. Will be pleased with his contribution.
10. Ashley Barnes 5.5
Fairly anonymous in the first half and produced a couple of wayward passes with Burnley on the front foot. Had a goal ruled out for offside in the first half, denied by Alisson in the second, and cleared Matip's header off the line in-between.
11. Chris Wood 5.5
Blew hot and cold throughout at Anfield. Made a nuisance of himself in the early exchanges, getting of a couple of deliveries behind Liverpool's defence, but couldn't compete with physicality of van Dijk and Matip. Forced Alisson into a first half save from McNeil's cross.