The Grecians took the lead thanks to a Harry Worley own goal, but Newport levelled a minute later through midfielder Lee Minshull.
Exeter are now three points above the relegation zone and both teams have now only won once in their last 14 matches.
After a goalless first half, the visitors stormed into the lead in the 50th minute when Newport defender Worley diverted Scot Bennett's driven cross into his own net.
However, Justin Edinburgh's men replied just a minute later. Grecians goalkeeper Artur Krysiak could not claim Ryan Burge's floated corner and the ball was eventually forced over the line by Minshull after a goalmouth scramble.