The hosts will have wondered how they were behind at half-time after dominating the first period but, for all their chances, they trailed to Conor Washington's 23rd-minute strike.
It came from a long throw on the Newport left, which Exeter failed to clear. The ball eventually fell at the feet of Washington, who lashed it past goalkeeper Artur Krysiak.
Exeter missed a great chance to equalise when David Wheeler's shot was well saved by Lenny Pidgeley and, with the goal at his mercy, Alan Gow headed against the post.
The second half was a different story with Newport looking the more likely to score and so it proved when Harry Worley found himself in space at the back post on 70 minutes and he lashed the ball into the roof of the net.
Scot Bennett had a late effort cleared off the line, but Exeter never looked like getting back into the game after a poor second period.