Aldershot nearly led before the break when Rhys Browne set up Lafayette but Marek Rodak saved and Charlie Walker failed to score on the rebound.
The visitors kept pressuring as Jake Gallagher and Jim Stevenson forced Rodak into further saves.
Victory for the Shots was sealed when Rodak brought down Lafayette and he scored from the spot on 76 minutes.
Aldershot Town boss Barry Smith told BBC Surrey:
Media playback is not supported on this device
"You know it was a very scrappy game and Welling gave us a lot of problems in the first-half.
"I thought at half time we upped our levels and we caused them problems in the second half and I think we deserved to win the game.
"I've got confidence in the strikers and confidence in players who can score and it was a matter of time. Their keeper made some terrific saves, but you've got to keep believing like we've done this season."
Ross Lafayette scored a second-half penalty to seal all three points for Aldershot Town as they beat Welling.