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Welling remain one point above the Vanarama Conference relegation zone after losing at Kidderminster on Wednesday night.
The game was shown live on BT Sport and the home side opened the scoring on 14 minutes when Jared Hodgkiss found the net from 20 yards.
Welling trailed 2-0 at half-time when a Chey Dunkley effort hit Jamal Fyfield and went in for an own goal.
Harry Beautyman halved the deficit eight minutes into the second half but the visitors, for whom player-manager Jamie Day completed 90 minutes, couldn't force an equaliser.
The Wings host fellow strugglers Nuneaton - now managed by ex-Welling boss Liam Daish - on Saturday.