Butler bundled in Tom Eaves' shot to give Yeovil victory in a match that appeared to be heading for a draw after an uninspiring encounter played in poor conditions.
Yeovil had an early chance to take the lead when Otis Khan's shot was deflected behind for a corner, before Darren Ward was forced to clear an effort from Newport's Butler.
The hosts grew into the game as the first half wore on, Matty Dolan shooting over after good work from Khan, before Ryan Hedges had a header well tipped over the bar by Joe Day.
Yeovil picked up where they left off at the start of the second half as Khan had a goalwards shot well blocked, but it was Newport who almost took the lead when Scot Bennett headed Ben Tozer's cross over from six yards out.
The game looked to be petering out into a draw, but Yeovil snatched a win in the 81st minute when Hedges found Eaves with a low cross and the substitute somehow bundled the ball home off the luckless Butler.