Liverpool enjoyed 35 shots at the goal of Burnley debutant Nick Pope but scored just once - an equaliser.
And it was required after Burnley scored with their only shot of the first half.
Liverpool's defensive failings were again exposed with this incident, but up front they just couldn't get the ball over the line. Headers went wide, shots went over and substitute Dominic Solanke even hit the cross-bar from about 4 yards out.
The reds put together some brilliant attacking moves but ultimately they drop another 2 points against average opposition and have just 8 points from their opening 5 games.
Klopp had made seven changes to the side which drew midweek against Sevilla - his focus he said was on ensuring players had the right intensity for the game.
But despite Liverpool's dominance Joel Matip was called into action late on to head off the line, after Mignolet failed to get near a Burnley header from a corner.
Mignolet, Klavan, Alexander-Arnold, Robertson, Milner, Coutinho, Sturridge all came into the side from that which opened against Sevilla. Only Emre Can, Joel Matip, Roberto Firmino and Salah kept their place.
Dejan Lovren was missing due to a 'back injury' however he was on the bench - suggesting that LFC may well have not done enough business in terms of centre-backs in the transfer window.
Within minutes of starting it looked like this would be tested. Burnley fired a ball into the box and Klavan and Matip both went for the same header. Chris Wood managed to get in the middle of the two Liverpool players and Klavan ended up on the ground while the game was stopped.
After short treatment he was cleared to continue.
In the 27th minute though Klavan and Matip hadn't learned their lesson and both went for the same ball - which bobbled across to Scott Arfield to smash past Mignolet.
Fortunately their lead didn't last long.
On the half hour, Emre Can's ball to Salah was met with the Egyptian's chest. He controlled the ball - spun around and fired in another goal to add to his burgeoning collection.
Chance after chance fell Liverpool's way - headers going narrowly wide. Nick Pope, on his first appearance in the Premier League, going on to make 8 saves. I can't remember the last Liverpool goalkeeper to make 8 saves in a game.
In the second half Coutinho came out of his shell and fired round the post, over the bar - you name it.
Klopp left his subs late, too late for some, but when Dominic Solanke came on he perhaps spurned the greatest chance of a game.
Salah appeared to have been fouled but a penalty appeal was waved away - the ball came over to Solanke inside the 6 yard box and whoosh - onto the crossbar. Klopp smiled ruefully on the sidelines - he mustn't have been able to believe his bad luck.
But despite all of Liverpool's dominance they let Burnley add to their 1 shot on target in the final 10 minutes. From a corner Matip headed off the line and then from the second corner Mignolet was forced into a front post reaction save.
Defeat on the basis of this performance would have been total humiliation so thankfully avoided!
It's not been a great week for the reds but they have to pick themselves up. At some point, somebody like Arsenal will be on the end of another Liverpool hiding but today Burnley got another great result to add to those they got against Spurs and Chelsea in what is shaping up to be a good season for them.
Man of the Match: Mohammed Salah