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Roberto Firmino smashed home from close range after Dean Henderson fumbled Trent Alexander-Arnold's rasping strike.
But before the Brazilian scored his second goal of the night, Fred produced a wayward pass while Luke Shaw carelessly lost possession.
The three individual errors saw the visitors bag their third goal at Old Trafford as they keep their Champions League qualification dreams alive.
Liverpool went onto beat Man United 4-2 in a huge result that sees them climb to fifth place in the league.
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