Results are presented for an initial survey of the Norma Arm gathered with the focusing hard X-ray telescope NuSTAR. The survey covers 0.2 deg$^2$ of sky area in the 3-79 keV range with a minimum and maximum raw depth of 15 ks and 135 ks, respectively. Besides a bright black-hole X-ray binary in outburst (4U 1630-47) and a new X-ray transient (NuSTAR J163433-473841), NuSTAR locates three sources from the Chandra survey of this region whose spectra are extended above 10 keV for the first time: CXOU J163329.5-473332, CXOU J163350.9-474638, and CXOU J163355.1-473804. Imaging, timing, and spectral data from a broad X-ray range (0.3-79 keV) are analyzed and interpreted with the aim of classifying these objects. CXOU J163329.5-473332 is either a cataclysmic variable or a faint low-mass X-ray binary. CXOU J163350.9-474638 varies in intensity on year-long timescales, and with no multi-wavelength counterpart, it could be a distant X-ray binary or possibly a magnetar. CXOU J163355.1-473804 features a helium-like iron line at 6.7 keV and is classified as a nearby cataclysmic variable. Additional surveys are planned for the Norma Arm and Galactic Center, and those NuSTAR observations will benefit from the lessons learned during this pilot study. ; Comment: Accepted for publication in the Astrophysical Journal, 10 pages, 4 figures, 3 tables