The Gills made all the early running and Vale survived a couple of scares before the much-travelled Burton, who will be 36 next month, opened the scoring in the 11th minute.
Debutant Robbie Findley, on loan from Nottingham Forest, chipped a cross from the right byline and Burton made no mistake with a six-yard header at the far post.
Burton and Findley proved too much for Vale's cumbersome defenders and Whelpdale cracked a second for the Gills in the 25th minute. The midfielder's run to the right of the penalty area went unchecked and he drove a 14-yard shot into the roof of the net from Danny Jackman's pass.
Sam Morsy, Tom Pope and Jennison Myrie-Williams all threatened as a shell-shocked Vale finally woke from their slumber as half-time approached.
Gillingham were content to protect their lead in the second half and did so in comfort, apart from a 68th-minute free-kick from Myrie-Williams which clipped the top of the crossbar.