The Magpies survived an early scare when a scramble in their box caused panic before Alan Smith was able to clear, and moments later Dean Morgan fired wide with an open goal at his disposal.
Tommy O'Sullivan was next to threaten for the Exiles as he blasted just wide of goal 12 minutes in, but on the quarter-hour mark Notts countered through a long-range shot from Graham Burke which was forced out for a corner.
Cooper's side was then denied a penalty in the 21st minute after Jon Stead appeared to be upended in the box, but the referee gave nothing.
Nevertheless the travelling fans would have been happy with how their side grew into the game and fashioned a number of other chances, culminating in Snijders getting a lucky break when the ball fell to him 12 yards out after Stead pressured opposition keeper Joe Day into a mistake, and duly sidefooting home.
Notts were bright after the restart, with Stead firing wide from Hewitt's cross 54 minutes in and Smith shooting from distance, but Newport then enjoyed a dangerous spell where both Haydn Hollis and Hewitt were forced to clear off the line.
A Burke free kick then flew just over the bar after the hour, while Stead also fashioned a chance which was saved by Day after a deflection.
It was end to end stuff for the remainder of the second half as Hollis once again cleared off the line and Burke and Stead weighed in with more chances, while Filip Valencic also came close and Adam Campbell struck the post late on.
Ultimately no more goals were to come, which suited Notts fine as they redeemed themselves in front of their manager and fans with a 1-0 win.
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