Liam Enver-Marum's sixth goal of the season has given Eastbourne Borough the advantage at half-time in a cagey affair at Park View Road.
Enver-Marum made the breakthrough in the 20th minute when he latched on to a through ball from Dan Smith and beat Charlie Mitten at the third attempt.
Borough have struggled to hold on to leads on a number of occasions this season and Welling should have equalised seven minutes later.
Ben Austin and Gary Elphick failed to clear a cross from deep but when the ball fell to Graeme Andrews he blazed his shot over from six yards.
Those have been the only two clear chances during a tight encounter where a lot of the play has been congested in midfield.
Enver-Marum glanced a header wide from Jamie Taylor's cross in the ninth minute and also had a fierce shot blocked by Mitten at the near post.
Welling, playing up a not inconsiderable slope, have enjoyed plenty of possession but their only other opening came when Loui Fazakerley headed wide.
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