The Blades went ahead on 24 minutes when Chris Porter nodded a long ball on to an unmarked Shaun Miller, who fired home sweetly on the volley.
In first-half stoppage time, a well-struck shot at the back post by Ryan Flynn was kept out by Magpies stopper Bartosz Bialkowski but, from the resulting corner, the Blades doubled their lead as Harry Maguire forced home Flynn's set-piece.
The visitors quickly put themselves out of sight in the second half with two goals in quick succession to leave County dazed.
They added their third on 52 minutes when a Flynn corner was again fired home with aplomb on the half-volley by Miller, before an attempted cross by centre-back Maguire dipped over Bialkowski two minutes later to make it 4-0.
Jordan Stewart curled home a consolation on 75 minutes with his first goal for the club but the game was long over as a contest.