Speed: Blades are ninth
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 after 25 minutes before Speed levelled four minutes later.
The former Newcastle and Bolton midfielder then wrapped up the three points as he lashed home from the penalty spot in the second half.
Kevin Blackwell handed a start to veteran defender Ugo Ehiogu after Chris Morgan's season came to a premature end following his dismissal against Hull last weekend.
City boss Gary Johnson gave recalls to David Noble, Nick Carle and Lee Trundle in a bid to turn around a run which has seen them go five games without a win on the road.
The Blades suffered a major setback after just three minutes when top scorer James Beattie limped off with what looked like a hamstring injury to be replaced by Rob Hulse.
This seemed to knock United out of their early stride and handed the impetus to the away side.
They nearly took full advantage after 11 minutes when Jamie McCombe hooked a volley just over the crossbar following Trundle's cross.
United responded well, though, and Matthew Kilgallon dragged a shot just wide after Adriano Basso had failed to deal with a deep corner.
But it was Bristol City who broke the deadlock when McIndoe lashed home from the spot.
Ehiogu clumsily brought down Dele Adebola just inside the area and the Scottish midfielder made no mistake as he hammered home into the left corner.
Ehiogu's woe was compounded immediately after the spot-kick when he was substituted for on-loan Manchester United midfielder Lee Martin.
United's response was as impressive as it was swift as they pulled themselves level just four minutes later.
Michael Tonge, who has been rejuvenated under the tutelage of Blackwell, arrowed a corner into the middle of the area and Speed was on hand to thump an unstoppable header into the bottom corner.
United started the second half in rampant fashion and should have gone ahead just a minute after the restart.
David Cotterill saw his low shot parried by Basso after a marauding burst towards the edge of the area and Hulse inexplicably prodded the rebound over the crossbar when it looked easier to score.
But United were handed a golden opportunity to take the lead after 54 minutes when McIndoe was adjudged to have handled in the area.
City's furious protests delayed the spot-kick but Speed showed immaculate composure to eventually rifle the ball into the right-hand corner.
City's response could have been immediate but Marvin Elliott conspired to prod just over the crossbar after Paddy Kenny had made a mess of a routine cross.
The Robins thought they had drawn level just after the hour but McIndoe was ruled offside after he glanced Trundle's cross past Paddy Kenny.
The pressure from City was relentless and Carle tested Kenny with a stinging drive from the edge of the area which the goalkeeper did well to palm away.
Kenny had to be at his best to deny Carle again with eight minutes remaining as he flew to his left to keep out the Australian's header.
There was no let-up from City and Trundle was the next to come close as his deflected shot skipped just past Kenny's post.
Johnson's side continued to press in the dying stages but they were unable to find a way past a stubborn Blades rearguard.