Argentinian midfielder Andres Gurrieri winged in with the first goal after 13 minutes before centre-back Curtis Nelson doubled the advantage and Lewis Alessandra headed home Luke Young's cross to cap a magical eight-minute spell.
Skrill Premier side Welling replied in the second half courtesy of Ross Lafayette's eighth goal of the season before Argyle shut up shop.
Boss Sheridan said: "We wanted a good start and the result, and that's what we got. We wanted to be in the third round draw, and we are, and that's the exciting part, especially for the supporters.
"It's pleasing that we got the three goals. When it's two you are always a bit worried but when you get three goals you have a good chance of winning, but obviously we never say that.
"However I think the players went out with that attitude in the second half and I am not too pleased about.
"I just thought we were very lazy in our game in the second half after such a bright start in the first.
"The next goal was going to be the important goal and they got it. They had one or two little chances where we might have got punished a little bit more.
"I don't want to sound too down because we had a lot of chances, especially in the first half, and we are in the hat and that is what everyone wanted from the game.
"The important thing is that we started well, played well in the first 20-odd minutes and got our three goals. You have got to give the team credit for that, and the fact there were three different scorers."
- Sports & Recreation
- Lewis Alessandra
- Curtis Nelson