Argyle, who beat Welling's Skrill Premier rivals Lincoln 5-0 in a first-round replay, needed just 13 minutes to take the lead.
Argentinian winger Andres Gurrieri latched on to a Reuben Reid pass before beating Wings left back Jake Gallagher on a meandering run and drilling the ball into the far corner past goalkeeper Lee Butcher.
Centre-back Curtis Nelson put Plymouth 2-0 up from 12 yards in the 18th minute as Welling failed to clear their lines following Argyle skipper Conor Hourihane's free-kick from the left touchline.
Butcher then saved well from Plymouth danger man Lewis Alessandra minutes later but the ball went for a corner, which was taken quickly and resulted in playmaker Luke Young sending in a pacy cross from the right which was headed home at the near post by Alessandra.
Welling responded when target man Ross Lafayette pounced on a slip by marker Neal Trotman to make it 3-1 in the 53rd minute, firing home from 15 yards, but it was not to be for the visitors.
- Sports & Recreation