The draw was York's fourth in succession, while a point for Exeter saw them pick up their second point of the season.
In a game of few clear-cut chances, the Grecians made a superb start and got the goal their early play deserved on eight minutes.
Matt Oakley picked up the ball on the left and delivered to the back post, where David Wheeler rose to head emphatically past York goalkeeper Michael Ingham.
York fashioned a good chance to equalise when Kevin Summerfield crossed, but the unmarked Keith Lowe headed wide from close range.
Graham Cummins missed a glorious chance to register his first Exeter goal when he cut out a poor pass, but shot straight at Ingham.
It proved costly as Lowe rose to head in Summerfield's corner with eight minutes remaining, just seconds after he had wasted an almost identical opportunity.
Arron Davies had a chance to win it for Exeter, as did Summerfield for York, but both players saw their efforts blocked.