Liverpool dropped more points at Anfield after Marcus Edwards' second-half equaliser was enough for Burnley to bag a 1-1 draw.
Dominik Szoboszlai missed a first-half penalty before Florian Wirtz looked to have put the Reds on their way to a comfortable afternoon.
However, Burnley kept themselves in the game and took their chance when it came through Edwards, who struck a clever finish past Alisson.
Liverpool finished the game having had 32 shots to an expected goals tally of 2.96 but never truly tested Martin Dubravka in the closing stages.
Slot's team have consistently struggled against the low block opponents have deployed against the defending Premier League champions and Burnley were the latest team to play the frustration game.
With Wirtz pulling the strings they should have had more than enough to overpower the visitors but, having reached the hour mark with only Wirtz's goals for their efforts, the momentum switched.
Alisson could not keep out Edwards' shot across him after Konate failed to close down the attacker on the left of the penalty area.
That undid all the work which had put them in total control.
Slot's team have slipped up at Anfield against Leeds, Nottingham Forest, Sunderland and Burnley already this season.
Liverpool boss Slot understands the frustration at full-time shown by Liverpool fans after their 1-1 draw with Burnley.
Slot's team have slipped up at Anfield against Leeds, Nottingham Forest, Sunderland and Burnley already this season.
"It's hugely frustrating," said Slot
"In football one team can have two chances and score one goal and another team can have multiple chances and score one goal. You then get the result we got today.
"In my head it wasn't just boos, it was frustration too. We are Liverpool and we're playing Burnley - we need to give them credit for the way they defended. But if we as Liverpool are not disappointed having a draw against Burnley then something is completely wrong.
"I and the players have the same frustration as the fans.
"We can talk about no loses in 12 but our standards are higher than that. Our standards are winning every game."
Burnley boss Scott Parker, whose side have not won in 13 matches - which includes nine defeats - praised his players for seizing their opportunity.
"We had to weather a lot in the first half but we showed more of our quality in the second half," he said.
"We changed a few things at half-time, it was important we were still in the game then, we were braver in possession and well worthy of our goal. It was an incredible finish.
"The penalty was soft but once the referee gives it on-field it's never going to be overturned. The key for us was getting in at half-time still in the game so we could fix a few things."
Burnley defender Kyle Walker on Soccer Saturday:
"You never say die. You go into the last kick of the game.
"Whatever will be will be, but at least we can look in the mirror and say we've given our all.
"This is a different challenge and one that I'm fully enjoying."
