Chris Wilder made two enforced changes as his team looked to make it three wins on the spin, goalkeeper Jordan Archer and Darren Carter in for the injured Matt Duke (thigh) and Kaid Mohamed (foot)
Newport manager Justin Edinburgh felt his side had been playing better than recent scorelines suggested and looked to have been provied right when County took a two-goal lead in the first half, Zebroski's 18th minute strike followed by Jones finding the net on 33.
Ivan Toney looked a threat for the Cobblers but his volley in between the home goals was well saved by goalkeeper Day.
Wilder made a double substitution at the break but a third goal on 53 minutes from Minshull stretched the hosts lead and looked to have made the game safe but goals from D'Ath on 76 and Richards on 85 made it a nervous last few minutes for County.