The goal came midway through the second half from Danny Webber, but by then City were struggling with Darren Purse already off injured and Rhys Weston a passenger in midfield.
The Bluebirds made just one change from the side beaten by Reading, with Weston taking over at full-back from the injured Neil Ardley.
The Blades were looking for the victory that could clinch Premiership football for next season and they used the flanks in the opening minutes to put the home goal under pressure.
City started to work themselves into the game and Steven Thompson fired over a couple of crosses that Paddy Kenny gathered safely.
Thompson was lively up front and he almost reached a Kevin Cooper cross before heading a Jason Koumas corner wide of the post midway through the half.
Two minutes before the break a run by Koumas put Cooper through but he sent his shot high over the bar.
By now City had already lost Willie Boland through injury and Jeff Whitley took his place in central midfield.
Purse failed to come out for the second half and Neil Cox took over in the centre of defence, while Ade Akinbiyi went up front in place of Neil Shipperley for United.
The impressive Thompson sent Koumas steaming through the middle but he scuffed his shot and Kenny made an easy save.
Steven Kabba joined Akinbiyi and Webber up front as the Blades went looking for the all important goal, but Glenn Loovens and Cox stood up well.
A mis-placed pass by Cameron Jerome in the 76th minute led to the Blades going ahead. The ball was worked down the left to Webber and his low shot curled in at the far post.
There were then loud appeals for a penalty as a Paul Parry shot struck Chris Morgan on the arm, but the referee waved play on and the Blades held on to take the points.