The visitors raced into the lead after just eight minutes through Taylor’s tap-in before Stockley struck the decisive second just before half-time, meaning Jon Stead’s second-half penalty was not enough to earn County a point.
It was the first league loss the Magpies have experienced in front of their own fans since losing to Portsmouth last April and Exeter were good value for their victory as they moved within a point of the automatic promotion places.
Their opener came when Elliott Hewitt impeded Stockley 25 yards from goal, with the free-kick only being half-cleared.
Dean Moxey unleashed a thunderous drive which cannoned off the post and straight to the feet of Taylor, who had the simplest of finishes.
Stockley then finished brilliantly from 20 yards after Taylor’s delightful cross-field ball.
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