A cruel late goal denied the Blades, who hit the woodwork four times in the live Friday night encounter.
Leon Clarke netted his 14th of the season and John Fleck was set off before a 91st minute winner for Bristol City.
Manager Wilder made three changes with goalkeeper Moore, fit-again Baldock and skipper Sharp returning to the starting line-up. Blackman, Carter-Vickers and Brooks dropped to the bench.
The first action was a Blades penalty appeal on two minutes but referee Coote did not appear well placed to see the clash between Smith and Duffy. Brownhill's early shot was comfortably collected by Moore and United put pressure on the visiting defence with Fleck's cross, kicked away at the back post by Flint, before Duffy stung the palms of goalkeeper Fielding.
Baker headed wide after a City short corner but United were on top in a fast-paced opening 30 minutes. Following Brownhill's foul on Stevens, Duffy's free-kick struck the back post before Clarke headed over after Baldock recycled possession.
Duffy shot at Fielding again before a Moore long clearance saw Clarke lay inside, allowing Sharp to strike first time - his effort clipping the top of the crossbar.
The Blades hit the woodwork for a third time on 38 minutes as City continued to survive United's onslaught. This time, from Fleck's corner, Sharp's header looked goalbound but Fielding pushed onto the crossbar and the visitors cleared.
After United's dominance, City taking the lead was a bitter pill with the last action of the first half. The visitors progressed to the edge of the box and PATERSON was given space to drill beyond Moore's right hand.
Far from being disheartened, United made an encouraging start to the second period - levelling with the first chance of the half. Lundstram's pass down the right channel saw CLARKE take his time before firing into a similar area to Paterson's goal.
United were attacking down both flanks and causing the visitors problems. On 54 minutes the home side struck the woodwork for a fourth time - Duffy's shot came back off a post and from the rebound Sharp's goalbound effort struck Baker.
The momentum was stifled, however, just before the hour when Fleck was sent off. As the City players surrounded the referee, he produced a red card for a tackle on Smith, who was replaced after treatment.
United refused to take a back step, despite the numerical disadvantage. Clarke attempted a half volley and a Baldock shot was on target but blocked. The visitors, though, saw more of a chance to attack and also had opportunities. On 67 minutes, Pack's heavily deflected shot was pushed away by Moore and Stearman blocked the rebound from Taylor. Brownhill also saw a shot deflected wide at the back post by Stevens.
Boss Wilder made two attacking changes, throwing on Brooks and Donaldson, followed for the final five minutes by Carruthers. One of the City substitutes, Leko, looked a threat but he fired wide after a couple of tricks.
A Blades break on 89 minutes caused nervousness for the visitors but Wright halted Carruthers with a timely tackle in the box and there was more drama - just as the officials indicated five minutes of added time. From Fielding's kick, the Blades half-cleared but Reid delivered to the back post and an unmarked FLINT stayed onside at the back post to beat Moore.
The remainder of injury-time saw City hold on to their lead professionally to claim the points - cruelly Blades fans would have though
Blades: Moore, Baldock, Stevens, Fleck, O'Connell, Basham (Carruthers 86), Lundstram, Clarke, Sharp (Donaldson 80), Stearman, Duffy (Brooks 76).
Subs: Wright, Carter-Vickers, Lafferty, Blackman.
Bristol City: Fielding, Wright, Flint, Baker, Bryan, Brownhill, Smith (Taylor 62), Pack (Leko 76), Eliasson (Djuric 45), Paterson, Reid.
Subs: Steele, Vyner, Magnusson, Woodrow.