Liverpool made seven changes to their starting line up, went a goal down and then mounted a resilient comeback.
With 12 wins and 3 draws after 15 games it is also now the best ever start for a top flight league season in the club's 126 year history, beating the 1978/79 and 1990/91 starts.
The reds dominated possession throughout the game, but the opening 20 minutes saw few touches for the temporary striking partnership of Daniel Sturridge and Divock Origi.
A first half injury to Joe Gomez forced a reshuffle and a scruffy Burnley opener in the second half placed the reds under pressure.
But Liverpool fought back quickly with goals from Milner and substitute Firmino turning the game on it's head. Xherdan Shaqiri netted in injury time to make the game secure.
Seven changes seemed a surprise from the Liverpool boss, however the vast majority of Premier League teams did make rotation the mid-week keyword. Even Burnley's teamsheet revealed five changes of their own.
Sadio Mane and Andy Robertson were expected absentees through injury. Dejan Lovren couldn't come in due to a clash of heads in training. But the front three were rested. Naby Keita coming in to midfield and Daniel Sturridge and Divock Origi given a chance to shine.
The reds made heavy work of the first half - and Burnley were keen to get in Liverpool's faces.
Although the reds were not significantly troubled, the reds took time to adjust to the pace of the game.
Joe Gomez was involved in a tackle that saw him receive treatment at the side of the pitch. A stretcher was called and Trent AA came on to take his place.
Liverpool settled towards the end of the first half and ended on the front foot.
But in the second half it was Burnley who scored from a scrappy corner when an unsighted Alisson could only palm the ball away for Jack Cork to tap in.
The reds response was fairly swift. A crossfield ball from Shaqiri and Naby Keita sailed into the area. He played it back out to the edge of the box where James Milner teed up a shot and drilled past Joe Hart.
With Firmino and Salah now sent on as substitutes, a quick second goal for the away team followed.
Alexander-Arnold floated a corner into the area, Virgil van Dijk played it across the face of goal and a simple tap in for Bobby Firmino.
Burnley never really looked like getting back into the game but they tried hard, a little too hard, and in injury time Alisson quickly released the ball and within seconds Salah had played it across to Shaqiri, in space, who had a simple finish.
Arms folded he stood in front of the Liverpool fans.
Another three points.
A gap is developing between the top two in the Premier League and the reds remain within two points of Man City.