The Robins have waited 17 years for a shot at the Championship play-offs and their fans were relishing every second of it before kick-off, even more so considering their home form has been one of the strongest in the division this season – bettered only by promoted Leeds United, Burnley and their play-off opponents.
But the Blades have been equally potent on the road and they thought they had the opening goal 10 minutes in, when a long throw was launched into the City penalty area and Campbell headed it into the net.
The away end was in full rapture before the assistant referee eventually raised his flag for offside against Sydie Peck, who was sharing a goalline tussle with Jason Knight.
Minutes later City almost made it 1-0 when Williams fired a rocket from 25 yards out that rattled the bar – an effort that would prove to be the only shot the hosts would have all night.
United finished 22 points ahead of the Robins in the table and they gradually proved their pre-match favourites status. They looked potent almost whenever they went forward, although initially failing to capitalise on their chances.
Rhian Brewster played Campbell into acres of space but in a one-on-one with Max O'Leary, the Robins keeper blocked the effort impressively with his feet. Brewster then broke behind City's defensive line and squared the ball, only for Moore to fire a tame effort at O'Leary.
It was another long through ball to Moore that led to the Blades' breakthrough, as Dickie was shown a straight red for hauling the Wales international down – his second dismissal this season against Sheffield United - and Burrows sent O'Leary the wrong way to score from the spot.