Since the Fermi discovery of γ-rays from novae, one of the biggest questions in the field has been how novae generate such high-energy emission. Shocks must be a fundamental ingredient. Six months of radio observations of the 2012 nova V5589 Sgr with the VLA and 15 weeks of X-ray observations with Swift/XRT show that the radio emission consisted of: 1) a shock-powered, non-thermal flare; and 2) weak thermal emission from 10 −5 M of freely expanding, photoionized ejecta. Absorption features in the optical spectrum and the peak optical brightness suggest that V5589 Sgr lies 4 kpc away (3.2-4.6 kpc). The shock-powered flare dominated the radio light curve at low frequencies before day 100. The spectral evolution of the radio flare, its high radio brightness temperature, the presence of unusually hard (kT x > 33 keV) X-rays, and the ratio of radio to X-ray flux near radio maximum all support the conclusions that the flare was shock-powered and non-thermal. Unlike most other novae with strong shock-powered radio emission, V5589 Sgr is not embedded in the wind of a redgiant companion. Based on the similar inclinations and optical line profiles of V5589 Sgr and V959 Mon, we propose that shocks in V5589 Sgr formed from collisions between a slow flow with an equatorial density enhancement and a subsequent faster flow. We speculate that the relatively high speed and low mass of the ejecta led to the unusual radio emission from V5589 Sgr, and perhaps also to the non-detection of γ-rays.