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The Wings were the better side for the opening hour but they were punished after failing to find a breakthrough while on top.
Harry Beautyman had the visitors' best first-half chance when a flowing move saw him link up well with Toby Ajala, only for him to fire the latter's cross wide of the post when it seemed easier to score.
Beautyman and Alife Mawson both went close for Welling, who nearly found a way through on the stroke of half-time but Joe Healy - at full stretch - was unable to turn the ball home.
After the break, Ross Lafayette headed straight at keeper James Belshaw before Healy's effort went over the bar.
Despite the pressure, Welling were undone in the 62nd minute when Wesley York was fouled and the resulting free-kick saw Andrew Brown's looping header beat returning keeper Lee Butcher.
The play-off chasing hosts wrapped up the points late on when Daniel Sleath's cross was headed in by substitute David Hibbert.
Welling are on the road again on Tuesday night when they visit Lincoln City.
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