Erik ten Hag's team were below par for the most part and were pegged back by an Oli McBurnie penalty after Scott McTominay's opener, but Diogo Dalot's exceptional long-range winner was enough to pile even more misery on the winless Blades.
The occasion was a particularly sombre one for Man United, who lost the legendary Sir Bobby Charlton at the age of 86 on Saturday morning, and a moving tribute to the 1966 World Cup was paid before kickoff at Bramall Lane.
Charlton's illustrious career saw him find the back of the net 249 times in a Man United strip, but in contrast, Ten Hag's side did not look like scoring in the early exchanges, where most of the action happened closer to Andre Onana's goal.
The Cameroonian was called into action in the 24th minute to deny Cameron Archer's low drive, although he failed to palm the ball away from the danger area and needed Dalot to cover his back.
Having withstood that early pressure, Man United went down the other end of the field and took the lead against the run of play in the 21st minute, thanks to another crucial contribution from Brentford hero McTominay.
The Scotsman firstly kept an overhit cross alive on the left and then received Bruno Fernandes's pass onto his chest, lofting the ball into the air and scuffing a volley, but Wes Foderingham was still left rooted to the spot as the ball trickled into the corner.
However, McTominay soon became the villain in his own penalty area just three minutes later, moving his arm towards a James McAtee cross and being penalised for handball, allowing McBurnie to level the scores with an unstoppable penalty.
Onana dived the right way, but McBurnie's strike was perfectly placed - clipping the inside of the post before nestling into the net - and the ball struck the woodwork again in the 44th minute, but unfortunately for Fernandes, his free kick rattled the bar before bouncing out.
Ten Hag's side ended the half strongly and spurned a glorious chance to restore their advantage in the third minute of added time, as Rashford's shot found its way through to an onside Rasmus Hojlund, but Foderingham managed to thwart the Dutchman from close range.
Paul Heckingbottom's half-time comments worked a treat for the hosts, though, as Onana was forced into two vital saves within the first six minutes of the second period, getting a fingertip to an Oliver Norwood drive before a quick change of direction to parry a long-range Rhian Brewster effort.
Hojlund and Rashford both spurned opportunities of their own during a frenetic period of end-to-end action, while Sofyan Amrabat also tried his luck from range in the 67th minute and witnessed his rocket strike the upright.
Dalot was the next to let fly from a similar position 10 minutes later, having been teed up by fellow full-back Victor Lindelof, and the Portuguese found the mark with a magnificent curling strike that Foderingham could only palm further into the top corner.
The hosts had little in the way of a late response as they remain bottom of the pile with just a single point to their name, while Ten Hag's team have gone above West Ham United into eighth place.
A Champions League showdown with Copenhagen at Old Trafford is next up for Man United on Tuesday, while Sheffield United travel to the Emirates for a daunting meeting with Arsenal in seven days' time.