Ched Evans put the home side ahead in the 10th minute, but after being much the best team in the first half, City were put on the back foot by the visitors in the second.
Norwich kicked off attacking the Barclay End goal and would have gone ahead early on if Paddy Kenny had not managed to keep out Darren Huckerby's shot with his legs.
Huckerby then delivered a couple of inviting crosses with eluded everyone. However, he had better luck in the 10th minute when his left-wing cross was deflected skywards then headed on by Jimmy Smith to Evans, whose shot from six yards found the back of the net after taking a bounce.
Martin Taylor released Jamie Cureton with a through ball, but his lob over Kenny also cleared the crossbar. The City striker came closer ten minutes from the break with a looping header from Mo Camara's left-wing cross.
Keith Gillespie fired wide at the start of the second half. Ten minutes into the half David Marshall saved David Carney's shot, then Kenny saved from Huckerby.
After Lee Croft had replaced Jimmy Smith, Matty Pattison fired a free-kick straight at Kenny.
Billy Sharp replaced Chris Armstrong and immediately fired wide from a tight angle.
Michael Tonge brought a save from Marshall and, as the visitors increased the pressure on the Norwich defence, Dion Dublin came on for Evans.
The veteran saw more action in his own penalty area than at the other end.
The Blades had the ball in the net but an offside flag cut short their celebrations.
Huckerby rounded Kenny but fired into the sidenetting from a tight angle, before the home fans were able to celebrate their team's fourth win in five games.