Left-back Moxey struck a crisp low drive in the 21st minute that Stuart Nelson was unable to get to, before Dobbie added a second after the break from a little further out.
Trialist Quincy Owusu-Abeyie scored a third after neat work from Dobbie as Palace continued their preparations for the new season in style.
The Premier League new boys took their time to get going and it was their League One opponents who had the best of the early chances, although their wayward finishing meant that Julian Speroni was rarely tested.
Debutant Jose Campana had an opportunity well saved by Nelson, but there was nothing that the goalkeeper could do about the game's opening goal.
A Palace corner was only half cleared and when the ball fell to the Moxey, he fired a 25-yard strike into the bottom corner.
Dobbie doubled Palace's lead midway through the second half when he picked the ball up just inside the Gillingham half before unleashing a low effort from 30 yards that found the corner.
Dobbie then turned provider a few moments later when he released substitute Owusu-Abeyie behind the Gillingham defence, with the former Arsenal and Spartak Moscow winger firing past Nelson to confirm the win.