After all the teams around them in the bottom half failed to win on Saturday, Paul Tisdale's side had the opportunity to move up to 16th place with a win in Wales.
A victory would have sent them five points clear of the relegation zone, but in the end they had to settle for a point as both sides' poor runs of form continued.
Exeter took the lead at Rodney Parade when Harry Worley put through his own net in the 51st minute, but the hosts' response was immediate as Lee Minshull scrambled home from a corner a minute later.
Neither side were able to force a winner, and they have each only managed to claim one win in their past 14 league outings.
In League One, Callum Wilson's stoppage-time goal rescued a point for Coventry City in a 2-2 draw with Port Vale at Sixfields.
It capped a remarkable comeback for the home side, who were two down with nine minutes remaining but recovered to earn a share of the spoils and frustrate their opponents.
Doug Loft gave Vale the dream start, chipping the ball over goalkeeper Joe Murphy before tapping home, and that lead was doubled 17 minutes from time when Ben Williamson nodded home from close range.
It looked as though Coventry were heading for a third consecutive defeat, but Mark Marshall crossed for Jordan Clarke to head home, before Wilson out-muscled the Vale defence to score for the fourth game in succession - his 18th league goal of an impressive season.
