County's task was not helped by Gary Liddle's 13th minute red card and United took advantage to go in front on the half-hour mark through Kevin McDonald.
Visiting substitute Enoch Showunmi had an immediate impact by scoring the equaliser on 59 minutes, but Harry Maguire's 67th minute header ensured it was a winning start for new Blades boss David Weir.
The Blades made a bright start with Jamie Murphy curling a free-kick straight at Bartosz Bialkowski and then firing wide after Mark Fotheringham had given away possession.
Just moments after Danny Haynes had come close for County, the away side were down to 10 men when Liddle was dismissed after his high boot caught McDonald in the ribs.
The hosts called for a penalty on 20 minutes after Chris Porter and Dean Leacock grappled in the box before taking the lead 10 minutes later when Darryl Westlake got down the right and pulled the ball back for McDonald to finish inside the post.
Weir's side continued to apply the pressure early in the second half as Bialkowski saved smartly from Michael Doyle and McDonald's swerving shot drifted wide.
Porter then had two good opportunities to increase the Blades' lead, but he firstly side-footed over following good work from Febian Brandy and then failed to apply a finishing touch to Neill Collins' shot.
And the Magpies made their hosts pay on 59 minutes when the ball broke kindly for Showunmi and he guided his 25-yard shot inside the upright.
But the Blades were back in front on 67 minutes when a deep free-kick found Maguire unmarked at the far post and the defender placed his header beyond Bialkowski.
New Blades signing Lyle Taylor came on as a substitute and saw a dipping free-kick pushed away by Bialkowski before Fotheringham's sweetly-struck volley whistled just wide at the opposite end.
In the closing moments, the Blades had two more chances to wrap up the win as Brandy was thwarted from close range by Bialkowski before on-loan Liverpool midfielder Conor Coady flicked an attempt wide, but those misses did not prove costly.