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Liverpool again needed stoppage time to secure victory as Hannibal Mejbri's handball gave Mohamed Salah the chance to strike a late spot-kick and hand Burnley a cruel 1-0 defeat.
With the Premier League champions without Alexander Isak, who needs more training time to get back to full fitness, Arne Slot's Premier League champions were heading to a wasteful draw.
They failed to score from 26 shots with around 80 per cent possession - until they were literally handed the win in stoppage time.
After Lesley Ugochukwu's red card for two bookings, former Manchester United midfielder Hannibal raised his hands needlessly in the area to block a cross, leaving referee Michael Oliver no choice but to point to the spot.
Salah, who did not have a shot for the entire game, stepped up to emphatically crash the ball into the top corner.
The goal kept Liverpool at the top of the table as the only team with a 100 per cent record - and Slot's side have extended their scoring record to 38 games.
Keane: Liverpool lacked quality - but that's what top teams do
Sky Sports' Roy Keane:
"They keep going.
They've got good options off the bench as we've found out in a few of their games.
"Burnley showed a lot of grit but Liverpool deserved to win the game by the way they dominated.
"Liverpool lacked quality, their final pass was so off today - but we were sitting there thinking they would get one more chance.
"And with Burnley, there's a bit of madness at the end just like at Old Trafford!
That's what the top teams do."
More to follow.
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