Burnley captain Ben Mee rose to head home a Robbie Brady corner in the 32nd minute as Sean Dyche's side secured a victory which moves them up to 16th place on 16 points, five points above the relegation zone.
Bottom placed United have just two points from their opening 16 games -- the worst ever start to a top flight season in English football.
The nearest Chris Wilder's side, who had suffered five squad members testing positive for COVID, came to an equaliser was a David McGoldrick shot in the 50th minute which was saved by Nick Pope at his near post.
United forward Rhian Brewster had a header cleared off the line by Josh Brownhill before Mee's goal, while the visiting goalkeeper Aaron Ramsdale had kept out an early effort from Ashley Barnes and a Chris Wood header.
The Yorkshire side enjoyed the majority of possession but struggled to break down a resilient Burnley defence, superbly marshalled by Mee.
The Clarets are now unbeaten in their last four home games and have battled the way out of the bottom three ahead of an impending takeover of the club by American company ALK Capital, which sources have indicated should be completed before their next game on Sunday.
Dyche, who said he had not been given information about the expected change in club ownership, praised his side's effort.
"Working hard without the ball has seen us through a tough game. We were playing against a Sheffield United side who I think are a good side, talk about a team that is fighting, they are fighting and put us under pressure in the second half," Dyche said.
The only bad news for Dyche was losing left-back Charlie Taylor to an early hamstring injury.
For Wilder, whose side finished ninth last season after winning promotion to the top flight, the goal of keeping the club in the Premier League looks increasingly remote.
"It has always been a fighting test right the way through. We had a perfect storm last year, it is the complete opposite this year," Wilder said.
"It has always been an incredible fight all the way through, it was a fight to get out of the Championship. The year we had last year was a perfect storm for us, we took big moments and found that bit of quality.
"This time we don't find that bit of quality and it marries up with the results. It is a big arm wrestle and a big struggle for us."