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.
Here are the ratings.
1. ARIJANET MURIC 4.5
Struggled to assert or impose himself on long throws, set-pieces and crosses into the box from open play. The Clarets needed their goalkeeper to be more commanding when threatened by Sheffield United's height advantage inside the penalty area. Denied Oli McBurnie three times in the second half while also denying Anel Ahmedhodzic from a corner. Photo: CameraSport - Andrew Kearns
2. VITINHO 4
Caught ball-watching when Iliman Ndiaye netted the first equaliser from close range, but combined well down the right hand side with Manuel Benson either side of the Blades making it 1-1. Made a few scurrying runs forward in the second half, though the home side's dominance allowed the Blades' joint leading scorer and Ben Osborn to exploit the space in behind. Photo: CameraSport - Andrew Kearns
3. TAYLOR HARWOOD-BELLIS 4.5
The Manchester City starlet met his match as Sheffield United's Oli McBurnie proved a handful. Stood in "No Man's Land" for the home side's second equaliser and unable to intervene as Jack Robinson converted the loose ball to give United the lead. Should have made it 2-0 when off-target with a free header, cleared McBurnie's touch off the line at 1-1, and tested Wes Foderinhgham with another chance from a corner when the game was gone. Photo: CameraSport - Andrew Kearns
4. JORDAN BEYER 4
The first time we've seen the Germany Under 21 international flustered as Sheffield United ran riot after the break. Afforded Anel Ahmedhodzic too much space inside the penalty area as the Bosnian guided a Jack Robinson cross into Oli McBurnie, and he seemed in disarray when the Bosnian defender made it 4-2. Also unaware of the danger that lurked to his left as George Baldock collected an Oli Norwood pass in the lead up to the fifth goal. Photo: CameraSport - Andrew Kearns