The visitors took a fourth minute lead when Whelpdale was given room to turn inside the box and lashed an unstoppable right-footed shot into the top corner from 15 yards.
Rovers battled back well to hit a post through John-Joe O'Toole before Burton produced a moment of quality a minute before the break, lifting the ball over his marker before burying a fierce volley from eight yards,
Rovers look a better unit under new boss John Ward but the Gills defended their lead with tenacity and skill after the break, restricting their hosts to few sights of goal.
Ward's men also looked secure at the back, despite the loss of skipper Garry Kenneth to injury just before half-time.
But Gillingham's extra quality showed in a game of precious few clear scoring opportunities.