The Reds' forgotten man Divock Origi was gifted an outrageous winner in the 96th minute at Anfield after Jordan Pickford's blunder.
Earlier, Pierre-Emerick Aubameyang scored two in Arsenal's thrilling topsy-turvy win over North London rivals Tottenham.
But it was a lot simpler for Chelsea in their all-West London affair with Fulham.
Liverpool 1 Everton 0
Arsenal 4 Tottenham 2
Chelsea 2 Fulham 0
Southampton 2 Manchester United 2
Crystal Palace 2 Burnley 0
Huddersfield 1 Brighton 2
Leicester 2 Watford 0
Manchester City 3 Bournemouth 1
Newcastle 0 West Ham 3
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