We present a new, probabilistic method for determining the systemic proper motions of Milky Way (MW) ultra-faint satellites in the Dark Energy Survey (DES). We utilize the superb photometry from the first public data release (DR1) of the DES to select candidate members and cross-match them with the proper motions from the Gaia DR2. We model the candidate members with a mixture model (satellite and MW) in spatial and proper motion space. This method does not require prior knowledge of satellite membership and can successfully determine the tangential motion of 13 DES satellites. With our method, we present measurements of the following satellites: Columba I, Eridanus III, Grus II, Phoenix II, Pictor I, Reticulum III, and Tucana IV. This is the first systemic proper motion measurement for several of these satellites, and the majority lack extensive spectroscopic follow-up studies. We compare these to the predictions of Large Magellanic Cloud satellites and the vast polar structure. With the high-precision DES photometry, we conclude that most of the newly identified member stars are very metal-poor ([Fe/H]<=-2), similar to other ultra-faint dwarf galaxies, while Reticulum III is likely more metal-rich. We also find potential members in the following satellites that might indicate their overall proper motion: Cetus II, Kim 2, and Horologium II. However, due to the small number of members in each satellite, spectroscopic follow-up observations are necessary to determine the systemic proper motion in these satellites.