The visitors held out until two minutes before half-time, when Carey's shot found the top corner past Joe Day.
In a frantic finale Newport substitute Zak Ansah saw his curled effort crash into the crossbar.
Carey then almost grabbed a second as the playmaker hit the Newport bar from a free-kick.
Newport County manager Terry Butcher told BBC Wales Sport: "We weren't good enough, nowhere near good enough.
"We never reached the standards that we've set in every match this season so far and paid the price.
"Had we got something at the end I felt it would have been a bit of a robbery job, although we did push four men forward at the end to try and get something."
Plymouth Argyle climbed to second in League Two thanks to Graham Carey's fourth goal of the season against Newport County, who are now bottom.