Despite taking the lead through Billy Sharp, United suffered a narrow pre-season loss at Burton Albion on Tuesday night.
The Brewers levelled before the interval through former Blade John Brayford, before Marcus Harness won it with the goal of the night in stoppage-time.
United were quickly out of the traps, McGoldrick drew a save from O'Hara inside two minutes, before Freeman hit the bar and Fleck struck the outside of the post within a minute of each other in the Blades next attack.
McGoldrick then almost capitalised on a mistake by Buxton but prodded an effort just wide of the far post.
Burton responded with a couple of efforts of their own. Firstly, Sbarra failed to connect clearly from Fraser's cross, before the home midfielder saw a shot comfortably saved by Moore from inside the box.
But on 18 minutes it was the Blades who deservedly went in front. Stevens fashioned space on the left for a near post cross and unsurprisingly it was SHARP who rose higher than his marker to head home past O'Hara.
On the half hour, Bryan produced a vital block to divert Daniel's dangerous low cross wide of the goal, before heading clear from the resulting corner. An unfortunate slip in possession almost cost the Blades the lead, but Moore was at his best to deny Akins, who saw a powerful shot tipped wide of the post.
Five minutes before the break, Burton did restore parity, with former Blade BRAYFORD heading home from Sbarra's inswinging corner and it was all square at the interval.
The Blades started the second period positively, McGoldrick was denied with a back post header, before Stevens saw his route to goal blocked. From the resulting corner, Stearman's header was tapped home by McGoldrick, but the assistant referee cut short the celebrations.
On 62 minutes, both sides made a number of changes, with United making a total of eight outfield substitutions. Robinson the replaced skipper Sharp, with both receiving a warm hand from the travelling Blades faithful.
Robinson looked lively from the off and his first touch as a Blade would have been a goal had it not been for a last-ditch block by Nartey. The former Preston man then got the better of Daniel and his low cross looked destined to be turned home by Clarke, until Hutchinson intervened.
With the clicking down, Robinson continued to look United's biggest threat, this time firing narrowly over from a promising position. Clarke was then sent free down the left, but Daniel again denied the Blades, with Robinson ready to pounce.
Two minutes of added time were signalled by the fourth official and it was the hosts who snatched the winner, with HARNESS scoring a goal that drew applause from all inside the Pirelli.
Burton: O'Hara, Brayford (Morley 88), Daniel, Buxton (Templeton 75), Wallace (Vale 79), Fraser (Hart 88), Harness, Akins, Sbarra (Hutchinson 68), Nartey (Anderson 81), Boyce (Dolan 88).
Blades: Moore, Baldock (Freeman 62), Stevens, Fleck (Lundstram 62), Basham (Jagielka 62), L. Freeman (Duffy 62), Sharp (Robinson 75), Norwood (Slater 62), McGoldrick (Clarke 62), Stearman (Egan 62), Bryan (O'Connell 62).
Unused: Dewhurst, York, Brunt, Boyes, Young.