However, a lack of communication between Boaz Myhill and Marc Joseph allowed Danny Adams to equalise 14 minutes later.
Half time substitute Michael Keane had the chance to put City back in front when Leon Cort won a debatable penalty, but County 'keeper Neil Cutler made a good save.
It didn't take much longer for City to restore the lead though as Jason Price scored a close-range header in the 61st minute, before Danny Allsopp wrapped up the win 4 minutes later courtesy of a defensive mix-up.
The three points sees City go top after Luton could only manage a draw at Peterborough.
City line-up (4-4-2): Myhill; Joseph, Cort, Delaney, Dawson; Price (Wiseman 90), Ashbee, Lewis, Edge (Keane 45); Allsopp (Facey 85), Wilbraham.
Un-used subs: Duke, Green.