Yorkshire rivals Leeds United lost their local derby 1-0 at Hull City, former Barnsley striker Jonathan Parkin scoring the only goal whilst on Friday night Watford were edged out 3-1 at Crystal Palace. Veteran Dougie Freedman cancelled out Marlon King's opener before Jordan Stewart and Jay DeMerit put past their own goal.
Neil Warnock had kept the starting line-up that beat Southampton 3-0 last weekend at Bramall Lane with Danny Webber and Neil Shipperley starting upfront.
United had to comeback from behind in this game, Josip Skoko took advantage of a poor Paddy Kenny punched clearance to head home with 15 minutes of time gone in the first half.
It may not have been pretty but United pressed and pressed and eventually they were rewarded with Webber's strike, which could prove crucial at the end of the season.
The first exchanges were tense but it was an open game from the very start, with a cross from Sheffield United's Chris Armstrong just hit with too much force for Neil Shipperley and Darel Russell firing over at the other end.
The Potters, who had manager Johan Boskamp back in the dug-out after illness, went much closer in the 10th minute when Junior's lobbed diagonal ball was touched past David Unsworth by Carl Hoefkens but the Belgian right-back struck wide of goal.
And in the 15th minute, City edged ahead largely thanks to Kenny.
The goalkeeper came all the way to edge of the area to punch Clint Hill's long free-kick but only cleared as far as Skoko, who gratefully headed it into an empty net.
The Blades were hanging on and seven minutes later, Mamady Sidibe out-muscled Chris Morgan but blasted well over Kenny's crossbar.
However, Paul Ifill did threaten for the visitors when Rob Kozluk's long throw fell to him on the edge of the box but the winger's powerful strike was held by City goalkeeper Steve Simonsen.
The action remained frantic but Skoko's strike still separated the teams at the interval.
Despite Blades boss Neil Warnock throwing on Ade Akinbiyi and Steve Kabba, the visitors were still struggling to impose themselves on the game. The Potters continued to offer a real threat with Sidibe giving Chris Short, who was clearly struggling, and Chris Morgan a torrid time.
The Mali striker volleyed wide from a clever ball from Luke Chadwick and stayed on his feet only to fire into the side-netting following a heavy challenge from Short.
Russell also went close for the hosts, just failing to reach a Sidibe lob and then getting involved in an ugly incident with Kenny and Short.
Michael Tonge had been at the centre of the visitors' best moments and he set up a chance for Webber before firing wide himself at the end of a fine run.
But the game hinged on a short spell with just 10 minutes left. City substitute Lewis Buxton broke down the right and crossed for Skoko, who had his low effort pushed onto the post by Kenny.
As the play went from end to end, Chadwick countered but was robbed by Short, whose long clearance was collected by Webber and he beat two defenders before sliding the ball past Simonsen.
Kabba then saw a volley fly just wide as the Blades pressed for a winner but the points were shared.