The Nuclear Spectroscopic Telescope Array (NuSTAR) serendipitous survey has already uncovered a large number of active galactic nuclei (AGNs), providing new information about the composition of the cosmic X-ray background. For AGNs off the Galactic plane, it has been possible to use existing X-ray archival data to improve source localizations, identify optical counterparts, and classify the AGNs with optical spectroscopy. However, near the Galactic plane, better X-ray positions are necessary to achieve optical or near-IR identifications due to the higher levels of source crowding. Thus, we have used observations with the Chandra X-ray Observatory to obtain the best possible X-ray positions. With eight observations, we have obtained coverage for 19 NuSTAR serendips within 12{deg} of the plane. One or two Chandra sources are detected within the error circle of 15 of the serendips, and we report on these sources and search for optical counterparts. For one source (NuSTAR J202421+3350.9), we obtained a new optical spectrum and detected the presence of hydrogen emission lines. The source is Galactic, and we argue that it is likely a cataclysmic variable. For the other sources, the Chandra positions will enable future classifications in order to place limits on faint Galactic populations, including high-mass X-ray binaries and magnetars.