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.
1. Nick Pope 6.5
Handling was clean and saved well at his near post to turn Salah's effort for a corner in the first half. Should have done better to keep out the Egyptian's disallowed finish, but found a way to divert Mane's awkward effort to safety prior to the Reds' second goal.
2. Matt Lowton 6
Indecision between himself and Gudmundsson afforded Tsmikas too much time and space to cross for the opener. Restricted openings down that side of the pitch in the first half otherwise, but far more stretched after the break as the hosts got into their stride.
3. Charlie Taylor 6
Made a number of decent interventions as the covering defender but eventually exposed by the wealth of talent on show. Dug in as best he could against the likes of Salah, Elliott and man of the match Alexander-Arnold, who created seven chances by getting in behind.
4. James Tarkowski 6
Almost equalised when beating Alisson to Lowton's set-piece in the first half and battled well on the whole, though questions will be raised about his positioning for Liverpool's second goal when failing to get to Alexander-Arnold in time to prevent the assist.