Neil Warnock's promotion favourites kept up their push for the Premiership as the Bramall Lane side missed a hatful of chances after taking the lead with a freak goal.
Ashley Ward and Jack Lester were the villains, both missing chances after Michael Tonge had broken the deadlock in the 44th minute.
Chris Armstrong, playing in an unaccustomed midfield role, picked out Lester in the 12th minute but the former Nottingham Forest striker tried to feed Ashley Ward instead of picking his spot and Arthur Gnohere cleared.
Burnley survived minutes later when Gnohere was caught flat-footed. The ball went through his legs to Peter Ndlovu and the United midfielder pumped in a curling cross, which Ward lunged at but could not get a final touch.
Burnley keeper Brian Jensen had luck on his side when Ndlovu lashed in a strike from the edge of the box. He failed to hold on to the shot and was challenged by Lester as the ball spun agonisingly the wrong side of the post.
The Clarets, who had won just one of their last eight games, were in trouble again in the 35th minute, as dangerman Ndlovu coolly teed up the ball before finishing with a volley that was snapped up on the line by Jensen.
Armstrong went close minutes before the interval as he charged into the area, his left-foot shot rising the wrong side of the bar. Burnley, outclassed for most of the half, caught the Blades on the break but Richard Chaplow wasted the chance heading Dean West's cross over the top.
The Lancashire side finally crumbled under the intense pressure in the 44th minute. Lester surged into the area as West and Gnohere converged on him. Gnohere stuck out a foot to hack the ball clear and it spun off West into the path of the unmarked Tonge who slotted it into an empty goal.
Ward and Lester should have increased the lead in the second half as United failed to take their chances.