The Clarets’ stretch last 16 games, though it looked as though it could have been prolonged as they led 2-1 at the interval in South Yorkshire.
Manuel Benson’s double, scored either side of Iliman Ndiaye’s equaliser, had given the visitors the advantage at the home of the Blades.
But Paul Heckingbotham’s side scored four second half goals to close the gap on the league leaders to three points while owning a game in hand.
Scotsman Oli McBurnie, a constant thorn in Burnley’s side, scored twice after the break while Jack Robinson and Anel Ahmedhodzic netted in-between.
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But Paul Heckingbotham’s side scored four second half goals to close the gap on the league leaders to three points while owning a game in hand.