A cruel late twist to this first leg ensures United will be playing catch up when the play-off sei-finals re-commence in 90 hours.
The Blades led through Kieron Freeman and looked the better side for long periods in this fixture.
But Sam Ricketts levelled for the visitors and Nathan Byrne scored a stoppage time winner past the right arm of Mark Howard, who earlier saved a penalty.
Blades fans will take no consolation in playing a major role in an entertaining fixture in front of the Sky cameras, whilst Nigel Clough will need to lift his team ahead of Monday's second leg.
As expected, Basham and Murphy started for the Blades. In total, gaffer Clough made six changes after rotating his squad for the final regular season game. Steven Davies, Holt, Flynn and Coutts also started, whilst Scougall, Adams, Doyle and McNulty dropped to the bench - Baxter (illness) and Reed (heel injury) missing out.
Basham began in midfield but it was Davies and Flynn who first caused problems for the visitors inside a Bramall Lane where the decibel levels had been noticeably turned up early on. It helped, undoubtedly, as the Blades made the better start.
With Holt also in a good position to shoot, Murphy scuffed wide after cutting in from the left and for Swindon, Gladwin beat Brayford but was well off target.
Murphy was prominent in United's early attacks and it was his cross that hit Gladwin's arm on 11 minutes but no penalty was forthcoming, despite huge appeals. Harris had two efforts in the opening exchanges, the first flew over the crossbar and had Davies got a touch to the second, the Blades would have taken the lead.
United claimed the advantage, however, soon after. On 19 minutes, from a move on the right, Flynn was persistent and at the second time of asking, his low cross found the late-arriving FREEMAN, who turned home from 12 yards, despite Foderingham getting a hand to the ball.
The visitors hit straight back and within a minute could have levelled but Hylton could not attach himself to Smith's ball across on the box, whilst on 27 minutes they were handed a route back into game when Harris fouled Byrne, referee Bond offering no hesitation before pointing to the spot. Gladwin's blast in front of the Kop was parried out by Howard and United, through McEveley, cleared up the danger.
There was no further punishment for Harris, but he was booked moments later for another foul on Byrne, whilst the man in the middle made himself more unpopular with the Blades fans when he said no when Murphy fell under a challenge from Stephens in the box.
In a high-tempo, entertaining game, from both sides, the referee got himself involved again when he booked Davies for kicking the ball away, a second after he was adjudged to have fouled.
The last major action of the half, on 40 minutes, saw Howard save with his legs from Byrne's shot after Hylton's run had caused problems for McEveley, but in a collision Brayford needed lengthy treatment.
Brayford hobbled through the remainder of the half but did not re-appear for the second period, Clough opting to introduce Scougall and switch Basham to central defence. The first chance after the break saw Smith head over Turnbull's cross but Swindon were soon level. Another cross, this time a Byrne corner, saw Ricketts rise above the defence to head into the top corner.
A Murphy run resulted in Freeman eyeing a second, but he slammed wide, whilst Swindon's start to the second half almost brought another for Ricketts, but his shot was deflected wide by Freeman.
Midway through the half, the Blades created a couple of half-chances. Basham's flicked header from Flynn's deflected cross landed in Foderingham's arms, whilst a good move involving Flynn, Coutts and Freeman resulted in Davies shooting wide.
On 70 minutes, from a Coutts pass, Flynn ran at the Swindon defence but the winger's shot was saved by Foderingham, whilst an attacking change saw Holt replaced by Done, who was quickly in the action, forcing the ball home but he was adjudged to have fouled Thompson in the build-up.
The referee riled the crowd again when he failed to play an advantage when Done was in the clear and a couple of chances put United back on the front foot. A Davies blast was pushed away by the visiting goalkeeper and the Blades striker's flicked header was inches wide from Murphy's cross on 78 minutes. Soon after Kasim gave away possession but Scougall could not turn in Done's near post cross.
With five minutes remaining a 40-yard run from Stephens resulted in him shooting into the side-netting from an acute angle, whilst the Blades were forced to sub a sub when Scougall limped off, replaced by Doyle.
All the Blades changes had been made when Davies picked up a knock late on but the attacking intent continued. Luongo made a great clearing header when Doyle looked like connecting with Flynn's cross. Turnbull was the next to make a vital block, keeping out Done.
United won a couple of corners in five minutes of added time but, cruelly, Swindon claimed a winner with little over 90 seconds of the injury-time period remaining. BYRNE was afforded space and his long-range effort bounced and spurted over Howard.
It was a cruel blow for the Blades, who played well for the majority of the fixture.
Blades: Howard, Harris, Brayford (Scougall 45, Doyle 88)), Basham, Flynn, Murphy, Coutts, McEveley, Freeman, S. Davies, Holt (Done 74).
Unused: Alcock, McNulty, Turner, Adams.
Swindon: Foderingham, N. Thompson, Byrne, Luongo, Stephens, Turnbull, Gladwin (Swift 68), Kasim, Smith, Hylton (Williams 76), Ricketts.
Unused: Obika, Barthram, Rodgers, Marshall, Belford