Tully pounced in the 54th minute before skipper Taylor doubled Exeter's lead to extend the Magpies' own winless sequence to six games in all competitions.
Alan Gow then fired over from distance and Ryan Harley got in behind the hosts' defence but ran of pitch before Nelson had to come sliding out of his area to prevent Exeter striker Cureton scoring early in the second half.
Ryan Harley then skipped through a stationary County defence to feed John O'Flynn and though the Irishman's shot was well saved, Tully fired home the rebound.
The dominant Grecians then got a deserved second when Ryan Harley's corner was headed powerfully home by Taylor.
County had to wait until the 90th minute for their best chance when Lee Hughes' hooked shot on the turn crashed against the woodwork.
Source: PA
Source: PA