Sean Dyche delighted with Burnley's defensive resilience
Ben Mee's 32nd-minute header from a Robbie Brady corner proved enough to see off the Blades, coming after Ashley Barnes and Chris Wood had forced two fine saves from keeper Aaron Ramsdale.
The Blades had an effort cleared off the line by Josh Brownhill, and were on the front foot throughout the second half, but couldn't find a way past the Clarets' miserly defence.
And Dyche said: "I thought we started the game well and took the game on.
We created two golden chances but they had a couple of good moments without really opening us up.
"Ben Mee scores a really good header from a great cross which was pleasing.
"Then all credit to them because in the second half they took the game on.
"I am a big fan of Chris (Wilder) and his team and people talk about them not winning a game, but they are fighting with their legs, energy and a bit of quality.
"I was pleased with our defensive performance as the game wore on and the way we saw the game through."
Dyche was pleased with how his side closed out the game, especially given the loss of Charlie Taylor to injury, and with a number of key players absent: "I am super pleased with that.
"Every time we get one back then we get more injuries which is a real challenge for us.
"That has happened again tonight.
"We are asking a lot of Erik Pieters and we put Josh Benson on who is a young central midfield player earning his spurs and he has to go wide and do a shift.
"So the defensive shape of the team was pleasing.
"They had a day's more rest. I can't see the value of teams playing two games in three days when there are no supporters in the stadium and no policing. I understand the media which puts a lot of money into football but equally there is a balance to the well-being of players and the fitness. It was tough for us on that script. It put a lot more pressure on the team tonight."