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.