The clustering of quasars on small scales yields fundamental constraints on models of quasar evolution and the buildup of supermassive black holes. This paper describes the first systematic survey to discover high-redshift binary quasars. Using color-selection and photometric redshift techniques, we searched 8142deg^2^ of Sloan Digital Sky Survey imaging data for binary quasar candidates, and confirmed them with follow-up spectroscopy. Our sample of 27 high-redshift binaries (24 of them new discoveries) at redshifts 2.9<z<4.3 with proper transverse separations 10kpc<R_T_<650kpc increases the number of such objects known by an order of magnitude. Eight members of this sample are very close pairs with R_T_<100kpc, and of these close systems four are at z>3.5. The completeness and efficiency of our well-defined selection algorithm are quantified using simulated photometry and we find that our sample is ~50% complete.