Billy Kee opened the scoring with a close range finish after Joe Day could only parry Matty Pearson's shot.
Josh Windass doubled the lead two minutes later when he converted Piero Mingoia's cross from the right.
Stanley move up to fourth but remain one point outside the automatic promotion spots.
Newport's third successive defeat sees them drop to 18th in the table, but they remain 13 points clear of the relegation zone.
Newport County manager Warren Feeney told BBC Radio Wales: "They are an honest bunch but today they let me down. I thought it was a below par performance after the goals.
"My son's eight and he would probably run around the pitch more enthusiastically than some of them.
"The aim was to stay up and get over the line and we're getting closer. The quicker we do it, the better."
Accrington Stanley kept pace with their League Two promotion rivals as two goals in two first-half minutes secured victory at Newport County.