We present the first results from the Swift Ultraviolet Survey of the Magellanic Clouds, the highest resolution ultraviolet (UV) survey of the Magellanic Clouds yet completed. In this paper, we focus on the Small Magellanic Cloud (SMC). When combined with multiwavelength optical and infrared observations, the three near-UV filters on the Swift Ultraviolet/Optical Telescope are conducive to measuring the shape of the dust extinction curve and the strength of the 2175{AA} dust bump. We divide the SMC into UV-detected star-forming regions and large 200-arcsec (58pc) pixels and then model the spectral energy distributions using a Markov-chain Monte Carlo method to constrain the ages, masses, and dust curve properties. We find that the majority of the SMC has a 2175{AA} dust bump, which is larger to the north-east and smaller to the south-west, and that the extinction curve is predominantly steeper than the Galactic curve. We also derive a star formation history and find evidence for peaks in the star formation rate at 6-10, 30-80, and 400Myr, the latter two of which are consistent with previous work.