The win for the Blades at Bramall Lane in front of a festive afternoon crowd of 20,538, also cut Charlton’s lead at the top of the division to seven points, with all the top sides now having played 22 matches.
But it was County who went ahead when Alan Judge’s 30-yard free-kick floated over the United defence and in. Former Falkirk player Ryan Flynn slotted home the equaliser before half-time when he turned in Richard Cresswell’s low, right-wing cross.
And the winner came early in what was a frantic second half when Magpies defender Alan Sheehan knocked the ball into his own net after tussling with Cresswell.
The Nottingham side finished the game with 10 men after substitute Ishmel Demontagnac was shown a red card in injury time by referee Jon Moss.