Even without skipper Graham Kavanagh, the cash-strapped Welsh club more than matched a United side who had held Arsenal to an FA Cup stalemate in midweek.
After a battling opening period, City upped the tempo in the second half and it was Joe Ledley who scored the all important winner.
City suffered a severe blow after just 13 minutes when Peter Thorne left the field to be replaced by Lee Bullock. The former York midfielder went upfront to partner Cameron Jerome.
Jerome showed plenty of endeavour backed by strong vocal support from the home fans, but apart from a cross from Richard Langley that was pushed on to his own bar by Blades keeper Paddy Kenny, there was little to suggest they could prise open the visitors defence.
United had the benefit of the strong wind and they sent numerous high balls into the City danger area to make use of that advantage, but new skipper Daniel Gabbidon and James Collins were strong at the heart of the defence.
City's first shot on target came in the 47th minute when a weak attempt by Bullock was easily gathered by Kenny, but the Bluebirds piled on the pressure and Kenny was forced to punch clear as Cardiff made use of the strong wind.
A cross by Chris Barker was met by Bullock but his glancing header drifted wide of the post.
City scored the goal their second-half performance deserved in the 75th minute and it was made by dangerman Jobi McAnuff.
He jinked and turned wide on the left before crossing low into the six-yard box. The ball was hacked clear but only to Ledley who thumped a left-foot shot into the corner of the net from 12 yards out.
Neil Warnock sent on Alan Quinn and Simon Francis in the closing minutes in an effort to rescue a point but City held firm.