We discuss the analysis of a series of 20 optical/near-IR spectra of the exoplanet host star 55 Cnc, obtained using a recently enabled spatial scanning mode with the Space Telescope Imaging Spectrograph (STIS), in order to assess the performance of that new observing mode for studies of transiting exoplanets. Standard pipeline reductions of the CCD spectral images were augmented by custom procedures for removing both cosmic rays and the strong fringing seen at wavelengths longer than about 7000 A. Both total (“white light”) fluxes and the fluxes for several narrower wavelength intervals were extracted from the processed spectral images. Apart from a slight (~400 ppm) offset between the relative fluxes in the two orbits sampled by the observations, the patterns exhibited by the ten flux values within each orbit are quite similar; the systematic differences in the fluxes are somewhat smaller than those seen for archival pointed/saturated STIS spectra of 55 Cnc. A parameterized de-trending method very similar to those commonly used to remove instrumental effects from time series observations of exoplanet host stars was then applied to the extracted fluxes. For the total fluxes, the scatter about the de-trending model is of order 30 ppm — comparable to the best precision previously obtained for time series photometry with HST. The scatter is somewhat larger for the narrower wavelength bins — particuarly at the longer wavelengths where the CCD is less sensitive; the de-fringing does reduce the scatter by 15-20% at the longer wavelengths, however. While these data are rather limited, they do suggest that spatial scanning with the STIS CCD can yield high-quality optical/near-IR spectra of the brighter exoplanet hosts.