After winning three shootouts, the Hellenic League side never looked in danger after skipper Moore swivelled inside the box to hook an emphatic shot beyond Sam Wilson after 14 minutes.
Long Eaton had opened brightly with Mitchell Robinson lofting over after a rare moment when he got in behind determined defender Tom Willment.
Then Danny Gordon hooked an effort on to the roof of the net after a long throw from Kieran Walker.
Yet Moore signalled his intentions with neat control and an instant angled blast after ten minutes that clipped the Long Eaton bar.
The hosts were lacking several key regulars through injury and suspension and rarely looked like getting back into the tie.
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