This result piles on the disappointment for Jurgen Klopp, whose side were frustrated in a similar manner by Sevilla in the Champions League on Wednesday, and were thrashed 5-0 by Manchester City last weekend.
Philippe Coutinho was back in the Liverpool team after failing to get his desired move to Barcelona and saw plenty of the ball in the early stages as Liverpool established the pattern of attack versus defence.
Burnley stood strong, but Liverpool's defensive frailties were highlighted as Chris Wood bullied the two centre-backs and poked the ball in the direction of Arfield, who finished confidently into the bottom corner on the half volley.
Three minutes later, the Reds were bailed out by their red-hot new signing. Emre Can spotted Salah's intelligent run from right to centre, and two great touches from the Egyptian teed him up to fire past Nick Pope.
Wave after wave of Liverpool attack hit the Burnley box for the remainder of the first half. Salah and Daniel Sturridge both tested Pope from the right edge of the box, with the Englishman also fooling the Kop into thinking the Reds had snuck in front with a shot that rustled the side-netting.
The second half began in similar style and Pope was kept busy. The Burnley stopper dived low to keep out piledrivers from Sturridge and Can, who scored in Liverpool's win in the corresponding fixture last season.
When Tom Heaton's stand-in was beaten, the woodwork came to Burnley's rescue. Trent Alexander-Arnold got to the right byline and crossed for substitute Dominic Solanke, who turned onto the crossbar from six yards on 85 minutes.
Burnley twice came close themselves in the closing stages through Ben Mee. From successive Robbie Brady corners, the defender had headers cleared off the line by Joel Matip and saved low by Simon Mignolet.
Mee, alongside defensive partner James Tarkowski and Pope, showed great resilience to withstand the Liverpool barrage as the visitors clung onto a point which keeps them sixth in the table, a place above Liverpool.