We report our multiwavelength study of the 2011 outburst evolution of the newly discovered black hole candidate X-ray binary Swift J1357.2-0933. We analysed the Swift X-ray Telescope and Ultraviolet/Optical Telescope (UVOT) data taken during the ˜7 months duration of the outburst. It displayed a 2-10 keV X-ray peak luminosity of ˜1035(D/1.5 kpc)2 erg s-1, which classifies the source as a very faint X-ray transient. We found that the X-ray spectrum at the peak was consistent with the source being in the hard state, but it softened with decreasing luminosity, a common behaviour of black holes returning to quiescence from the hard state. The correlations between the simultaneous X-ray and ultraviolet/optical data suggest a system with a black hole accreting from a viscous disc, and we do not detect X-ray reprocessing on the disc surface. The UVOT filters provide the opportunity to study these correlations up to ultraviolet wavelengths, a regime so far unexplored. If the black hole nature is confirmed, Swift J1357.2-0933 would be one of the very few established black hole very-faint X-ray transients.