Danny Webber netted the vital goal against Cardiff to move the Blades to within a point of securing the Coca-Cola Championship's second automatic promotion spot with three games remaining.
For Cardiff, another season in the Championship was confirmed as the play-offs drifted out of reach.
For most of the game it looked as though the visitors might be forced to wait beyond the Easter weekend to seal a top-flight return after 12 years in exile, but boss Neil Warnock's adventurous substitutions paid dividends.
Both sides made only one change to their starting line-ups, Cardiff handing Rhys Weston his first start since January in place of the injured Neal Ardley, and United drafting in Nick Montgomery for Paul Ifill in midfield.
Cardiff are one of the few teams in the division who can match Warnock's men physically but it was good football which saw them create the game's first chance, Steve Thompson cutting the ball back to Willie Boland who screwed his shot wide.
Neither side could gain the upper hand, Montgomery's looping header from David Unsworth's cross not proving in any way threatening.
Montgomery was a useful outlet on the right flank, helping put together the game's best move before hitting his finish straight into the ground.
Darren Purse had Cardiff's first real opportunity from a corner, but the Bluebirds captain failed to connect properly with his header.
Cardiff suffered a blow when Boland was forced off injured, Jeff Whitley his replacement.
The half appeared to be petering out until, in the final minute, Kevin Cooper was set free by Jason Koumas but blazed over the bar with the goal at his mercy.
Both sides made changes at the break, Neil Cox replacing Purse for Cardiff, and United swapping Neil Shipperley for Ade Akinbiyi.
The poor finishing continued as Cameron Jerome, who United tried to buy earlier this season, rifled over.
The excellent Thompson and Koumas looked threatening, and the latter's wicked inswinging cross was well smothered by Paddy Kenny.
United were struggling to contain the on-loan West Brom man and Chris Morgan resorted to foul means 25 yards out.
Cooper continued to be wayward when finishing, while at the other end Webber's header lacked both power and direction.
Chris Armstrong should have at least hit the target following good hold-up play from Akinbiyi, and it was the midfielder's final contribution before being replaced by Steve Kabba.
The front three looked to be having an effect and Webber should have opened the scoring but scooped Kabba's cutback over following a great pass by Rob Kozluk.
Webber made amends though. After being put through by Jerome's slack pass, he curled a shot wide of Neil Alexander and into the bottom right corner.
Both sides survived handball shouts in the box and Kenny set nerves jangling in stoppage-time when he spilt a catch in the box but United held on.