Buxton took the lead through Dom Tear at the Silverlands Stadium on 29 minutes, before second-half goals from Curtis Jones and Joe Duckworth saw FC United claim the three points and move to eighth in the Premier Division table.
Wilson joined the club ahead of the fixture on a month's loan from Fleetwood Town, and started in the tie.
The 17-year-old signed a new three-year contract with the Football League One side back in March, and played a part in both this year's and last season’s FA Youth Cup runs.
He has also been on international camps with Northern Ireland’s Under-17s and Under-19s.
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