It was the Shrimps' fourth straight win in League Two and they showed plenty of resolve after Justin Edinburgh's Exiles twice threw away the lead at home.
Billy Jones had struck on his Newport debut after joining on a short-term deal from Cheltenham on Thursday.
Conor Washington's cross found the new signing at the back post and he fired a superb low drive into the bottom corner after just three minutes.
The visitors equalised nine minutes later, though, when a quick counter-attack ended with Naylor turning Kevin Ellison's header into his own net.
Newport got back in front through Robbie Willmott, the winger smashing a fierce shot in off the underside of the crossbar from the edge of the area.
However, early in the second half, a terrible mix-up between Naylor and his goalkeeper Lenny Pidgeley saw the defender head Tony Diagne's hopeful ball into his own net.
Naylor's misery was then compounded 21 minutes from time as he brought down Ellison and Padraig Amond calmly slotted the penalty home.