Bristol's manager Gary Johnson had warned "Sheffield United don't look to pass the ball but to shove it down your throat" and their first-half power play was all about that.
James Beattie robbed City's keeper Adriano Basso with the ferocity packed into one shot which the Brazilian managed to parry for a corner.
He was soon diving to field a Beattie free-kick hammered from 30 yards and then found luck on his side when the United striker burst past the back four to drive in another shot.
Too many players had the wrong studs for a slippery autumn surface and the worst slip saw City exposed when their centre-back Jamie McCombe went sprawling.
Gavin Williams raced 20 yards to the rescue, thumping the ball out for another Blades corner.
City's attack focused too much on Dele Adebola's physical battle with Ugo Ehiogu. It was always a no quarter given affair with the striker lacking the support to be a real threat.
He should have scored with a header as Michael McIndoe lobbed to the far post, but sent it off target.
With the game at last showing purpose after the break, Paddy Kenny, in Sheffield's goal, made a double save from Lee Johnson and Lee Trundle.
On the hour Adebola made way for Stern John, who has arrived on loan from Southampton. Almost at once his downward header was bundled in by McIndoe, but he was ruled offside.
Basso made a great save to deflect a piledriver of a free-kick by Beattie as City kept their fourth clean sheet in a row.