German, on loan from Brentford, pounced on Argyle's failure to clear Bradley Dack's free-kick to fire the Gills ahead from close range after 31 minutes.
Whelpdale doubled their advantage shortly after the hour, tapping home when German could only hit the post having rounded Argyle goalkeeper Jake Cole.
Jason Banton's curling strike 14 minutes from time gave the visitors hope, but turned out to be a consolation for Argyle, who are now bottom of the Football League.
The hosts had lost three of their previous six games at the Priestfield Stadium, but never looked like dropping points against the relegation-threatened Devon side.
Whelpdale should have given them the lead inside five minutes, but wastefully headed over when unmarked in the area following Dack's cross.
Dack, Matt Fish and Danny Kedwell all had chances to add to the scoreline in between German and Whelpdale's goals.
Before scoring, Banton had the visitors best chance as he forced Stuart Nelson into a fine save to keep out his deflected, looping effort.
- Sports & Recreation