Ryan Williams was named in the starting line-up, making his return after having his tonsils removed, while Paul Musselwhite was back in goal after loan 'keeper Michel Kuipers was promoted to the first team.
After Blades forward Billy Sharp had hit the post early on, Daniel Webb gave City the lead on 8 minutes, heading in Carl Regan's right-wing cross.
But 20 minutes later, Ashley Ward equalised when he met an Ian Ross free kick.
The visitors continued to press for the winner, and it came 13 minutes into the second half. Adrian Harper - on as a substitute - got the ball and fired past Paul Musselwhite from 25 yards out to bag a 2-1 win.
City line-up: Musselwhite; Wiseman, Holt, Burton, Peat; Regan, Melton, Fry, Williams (O'Neil 74); Webb, Forrester.
Un-used subs: Harvey, Russell, Byron, Cooper.