The Clarets kept a sixth clean sheet of the campaign as skipper Ben Mee's first half header proved to be enough to blunt the Blades at Turf Moor.
Mee rose above Ethan Ampadu to head Robbie Brady's corner beyond Aaron Ramsdale, who had already made a couple of outstanding saves to keep out Ashley Barnes and Chris Wood.
Burnley defended doggedly to preserve their slender lead, though Nick Pope was only called into action when saving David McGoldrick's strike.
The home side, who have clocked up 11 points from their last five games, climbed above Brighton in the Premier League while Sheffield United were left without a win in 16 games.
1. Nick Pope 7
Unfortunate not to have had clean sheets in the previous two games, but finally got what he deserved. Eased the pressure with a number of high claims inside the penalty area and made a big save to deny McGoldrick in the second half. Photo: Nigel French
2. Matt Lowton 7
Kept things nice and compact, forcing Wilder to switch things about as a result. Tracked the runs of Robinson, Osborn and Stevens extremely well, stood his ground confidently, and limited United's productivity down that side of the pitch. Photo: James Gill
3. Charlie Taylor N/A
Pulled up holding his hamstring after a little shove from McGoldrick and was replaced by Benson in the ninth minute. Photo: James Williamson
4. James Tarkowski 7
Another commanding display from the Burnley centre back, who was on the end of everything that the away side whipped into the box. Dominated aerial battles - from crosses, set-pieces and long balls forward - and didn't give the strikers a sniff. Photo: Alex Pantling