Huub Stevens's men soon found their momentum, and went into the lead just after the half-hour mark through Greek defender Papadopoulos.
Augsburg had the chance to draw level on the stroke of half time when Paul Verhaegh found space inside the penalty area, but could only drag his shot wide.
Schalke took the game by the throat in the second half, and double their lead shortly after the break through United States midfielder Jones.
Jones was unlucky not to pick up his second on 68 minutes when Simon Jentzsch tipped his long-range strike onto the post, but Stevens's side didn't have long to wait for their third.
Dutch international Huntelaar put the game beyond Augsburg on 72 minutes with a clinical finish to make it 3-0.
Auguburg were outclassed, but they did manage to pull a goal back on 79 minutes courtesy of striker Torsten Oehrl, but it proved to be no more than consolation.