The Blades are without a win since the opening day of the season and sacked David Weir on Friday after just 13 games in charge.
The Sky Blues were the better side and doubled the lead four minutes before the break. Captain Carl Baker showed great skill down the right to get to the byline before pulling back for Callum Wilson to smash in his 10th of the season.
Leon Clarke added a third in the 49th minute, stroking into the bottom corner with his left foot after being teed up by Franck Moussa.
But United showed great spirit to pull one back with 28 minutes to play when substitute Lyle Taylor powered a free-kick through the wall.
Taylor headed in another 10 minutes from time to set up a tense finish but City held on to leave the Blades bottom of the table.