The Robins’ disappointment at defeat at Bramall Lane will be tempered by the fact that their own play-off spot was secured by virtue of other results going in their favour on Saturday.
City took the lead through Michael McIndoe’s penalty on 25 minutes, awarded after Ugo Ehiogu clumsily brought down Dele Adebola just inside the area.
However, Speed levelled four minutes later when he headed home Michael Tonge’s corner.
And the former Newcastle and Bolton midfielder then wrapped up the three points as he lashed home from the penalty spot in the second half after McIndoe was adjudged to have handled.
City are fourth, while the Blades are ninth, two points behind sixth-placed Crystal Palace.