Last year's beaten finalists Newport County fell at the first hurdle in the FA Trophy when they crashed at Park View Road.
Winger Joe Healy gave Conference South team Welling the lead with a fine header from 12 yards.
This came after Newport's Michael Smith had a similar chance saved by Sam Mott in the opening minutes.
Ben Martin scored in the second half to consign Justin Edinburgh's team to their fourth defeat in five matches.
VIEW FROM THE DRESSING ROOMS
Newport County manager Justin Edinburgh told Radio Wales:
"They [the players] were well short and bereft of any real know-how or determination or effort to win the game today.
"We've had some bad results of late, but we've always performed. But today we certainly haven't and we've got what we deserved.
"There's a lot missing, and that's application. In every game we've won this year we've earned the right to do that. And we are not good enough to just turn up and win and if we feel that's the case then what will happen is what happened today.
"This is a good group, but we've let ourselves down today and we're disappointed with it."