The second half began in similar style and the Burnley goalkeeper was kept busy.
Liverpool's weaknesses in defence are becoming a glaring issue for Klopp who admitted that he was aware of that fact, given the domination his team had enjoyed over Burnley in terms of possession and shots.
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.
"Yes, result wise it is frustrating but honestly I am as a person not like this", said Klopp, who made seven changes from the Sevilla game. Three minutes later, it was Joel Matip's turn to avoid the Liverpool goal again conceded from Ben Mee's hard header.
One mazy Joel Matip run nearly did the trick, but when the centre-back slipped in Salah, the Egyptian was tripped by James Tarkowski.
The 25-year-old's willingness to apply the dark arts to Burnley's game - time-wasting with his goal kicks after barely 15 minutes on the clock - was vital to the Clarets earning a point.
Burnley, meanwhile, were also arguably without their best player.
After short treatment he was cleared to continue. The ball was shot by Scott Arfield and nested in the corner of the Mignolet's goal. He controlled the ball - spun around and fired in another goal to add to his burgeoning collection.
They were unlucky, perhaps, not to see a late flurry of chances produce a winning goal, but by the final whistle, Anfield regulars were left asking the usual questions.
Philippe Coutinho and Alex Oxlade-Chamberlain could both start this week for Liverpool against Burnley.
He approached East after the match, and said: "I'm not too happy with the situation when Dominic Solanke hit the bar".
"If Salah goes down. if the centre-half goes down in the 18-yard box behind the striker, and if the striker goes to the ground". Klopp smiled ruefully on the sidelines - he mustn't have been able to believe his bad luck.
Defeat on the basis of this performance would have been total humiliation so thankfully avoided! The Reds will be buoyed by Coutinho's return and they will be eager to maintain their 100% home record.
Having appeared to announce itself as a potential title contender with a 4-0 victory over Arsenal in late August, Liverpool has slipped to eighth on the table, and Klopp will be concerned as to how his side will cope without Mane at Leicester in a week.