Unsuccessful in their last seven playoff campaigns and trailing 2-1 from the first leg, the Blades travelled to Swindon with the scoreline and the omens against them.
An already mountainous task grew significantly steeper early on as Nigel Clough's charges endured a disappointing first 10 minutes in which Town netted twice.
The assault continued as Massimo Luongo then went close, before Smith added a third on 18 minutes to make it 5-1 on aggregate and effectively end the contest.
After being aggrieved over the way in which they lost the first leg, Clough will have felt that his side were due a degree of luck, and it came moments after Smith's goal as Nathan Thompson deflected Jamie Murphy's shot past Wes Foderingham to give the visitors the faintest of lifelines.
Chris Basham raised pulses inside the County Ground by grabbing another for United, who went into the break trailing 5-3 on aggregate and knowing that a minor miracle was still needed in order to reach the showpiece on May 24.
It seemed unlikely that both sides would be able to sustain the thrilling tempo that made the first half so enthralling, but Swindon, despite their aggregate lead, refused to rest on their laurels and moved further ahead shortly before the hour mark.
Howard fouled Jermaine Hylton in the area, and Smith made no mistake from 12 yards, but Clough's side yet again refused to stay down and Ben Davies - who replaced Mark McNulty minutes earlier - saw his bullet header restore the visitors' slight hopes with 20 minutes to play.
If Sheffield's hopes were not already dashed, then they were on 84 minutes when Jonathan Obika added a fifth but, in keeping with the theme of a see-saw night, a late salvo from Matt Done and Che Adams saw Clough's men level up at 5-5 on the night.
Despite being within one goal of salvation, time ran out for the brave Blades on a night that saw their playoff heartbreak prolonged.
Swindon will face Preston North End in the final on May 24.