Ben Mee's 32nd minute header powered the Clarets to their third win in the last five games as they move up to 16th in the Premier League table, five points above the relegation zone.
Sheffield United remains at the foot of the table as they have made the worst start of any top-flight club ever, now winless in 16 matches to begin the season, equalling Queens Park Rangers' 2012-13 run.
Story of the match
The Blades had their first attack on the half hour mark as Mee scuppered David McGoldrick's ball behind for a corner and Josh Brownhill cleared Rhian Brewster's subsequent header with the rebound fired over.
Two minutes later, the hosts were in front. Robbie Brady's inch-perfect cross to Wood forced another spectacular save from Ramsdale and from the ensuing corner, Mee headed home to give the Clarets a 1-0 lead.
The Blades had a quickfire response, Mee heading away Enda Stevens' cross, but Brewster picked it up, took one touch and fired low that Nick Pope was able to save.
Brady nearly grabbed a second for Burnley, arrowing a free kick just over before halftime as the hosts took a slender lead into the dressing room.
In the 51st minute, Baldock played it inside for Stevens, who laid in a perfect pass for McGoldrick, who fired from a tight angle that Pope saved.
Sheffield United was seeing most of the ball in the second half, but a cross from Josh Benson took a deflection and Ramsdale had to be alert to tip behind.
Stevens and Ben Osborn tested Pope, who made simple saves and with 15 minutes left, Baldock shot and saw it take a deflection. The ensuing corner was cleared by Lowton.
Egan then just fired over, Osborn forced Lowton to again clear and McGoldrick also went over as the visitors were clearly having the better of the play, but had nothing to show for it.
With six minutes left, Pope saw Oliver Norwood's free kick headed out of his hands by his own player and Osborn, waiting at the back post, heads it into the six-yard area where the Clarets netminder scrambles back to collect.
That was the last meaningful action of the game as Burnley have taken 10 points from their last five matches while Sheffield United look for all the world as though they'll be an EFL Championship outfit next season.
Man of the match: Ben Mee
He not only scored a vital goal, but was rock-solid at the back where he led the Burnley defense as they soaked up and dealt with a bunch of pressure from Sheffield United.
Turf Moor is becoming a fortress again for Burnley
The Clarets are now unbeaten in five matches at home and Barnes and Wood up top are part of the reason. Mee and his defense partner James Tarkowski anchor the Burnley back-line while Pope has been outstanding in goal in that stretch.
Rhian Brewster has been a massive disappointment so far
Granted, he's only been in the starting lineup with any consistency in recent games, but he's low on confidence, not converting the chances he does have and has yet to score in a Sheffield United shirt. He deserves more time to show his quality, but he needs to get going soon if the Blades have any hope of staying up.