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Having been fouled by Alisson for the spot kick, Barnes fired home from 12 yards out to secure his side three welcome - and unexpected - points to move them away from the relegation zone.
Defeat means Liverpool remains six points behind leader Manchester United in the table, the stuttering champion enduring a fourth successive league game without a goal despite having 27 attempts.
Divock Origi hit the crossbar in the first half while Nick Pope - who starred when Burnley end the Reds' 100 per cent home record in their title-winning campaign - was outstanding in the visiting goal, helping secure a stunning result that has ramifications at both ends of the table.
