Ashley Barnes's winner in the 83rd minute was the striker's 100th career goal.
The 31-year-old picked himself up off the turf after being upended by Alisson to beat the Brazilian goalkeeper from the penalty spot.
And in doing so the match-winner secured Burnley' s first win at Anfield since 1974.
Here are our RATINGS from a memorable night in Merseyside...
1. Nick Pope 9
Jurgen Klopp described it as "Liverpool versus Nick Pope" last season and the Reds boss will be ruing the keeper's excellence again. His divine intervention frustrated the hosts as he made outstanding saves to deny Alexander-Arnold and Salah. Handling was exceptional.
2. Matt Lowton 8
The full back is beginning to show consistency in his performances. He pressed the ball well, giving Robertson and Mane very little room to work in. Supplied the header that allowed Barnes to turn the defence and win the penalty. Photo: Clive Brunskill
3. Charlie Taylor 6
Had plenty to contend with given Shaqiri, Oxlade-Chamberlain and Alexander-Arnold all took up those wide areas and made a splendid block to deny the latter in the first half. Replaced early in the second half after picking up an injury. Photo: Jon Super
4. James Tarkowski 9
Outstanding. A rock at the heart of the defence from start to finish. Eased the pressure with countless clearances, getting in positions to get a head or foot on the ball, and protected the penalty area heroically. Gave Liverpool's forwards very little.