We obtained medium-resolution spectra of 336 quasar candidates in the COSMOS HST Treasury field using the MMT 6.5 m telescope and the Hectospec multiobject spectrograph. Candidates were drawn from the Sloan Digital Sky Survey (SDSS) DR1 catalog using quasar flags set by the SDSS multicolor quasar target selection algorithm. In this paper we present our discovery spectra from 1.39 deg2 (69.5% of the COSMOS field) and a discussion of the selection method and yields. We confirmed 95 quasars, including at least two BAL quasars; 80 of these are new quasars that do not appear in previous quasar confirmation follow-up studies. The candidates additionally included 184 compact emission-line galaxies, a third of which are likely type 2 AGNs, and 12 stars. The quasars span a range in magnitude of 18.3<g<22.5 and a range in redshift of 0.2<z<2.3. Our results are consistent with a lower limit quasar surface density from SDSS color selection of 102 deg-2 down to g=22.5 over the entire COSMOS field. This work is the first step toward the eventual goal of setting up a grid of quasar absorption line probes of the 2 deg2 field and of conducting a complete census of supermassive black holes in this well-studied survey region. The total quasar count at the conclusion of this study is 139, making COSMOS one of the most densely sampled regions of sky where a grid of quasar sight lines can be used to probe the intervening volume.