We present the first high-energy X-ray (>10 keV) observations of the non-thermal filament G359.97-0.038 using the Nuclear Spectroscopic Telescope Array (NuSTAR). This filament is one of approximately 20 X-ray filaments of unknown origin located in the central 20 pc region in the Galactic Center near Sgr A*. Its NuSTAR and Chandra broadband spectrum is characterized by a single power law with Γ = 1.3 ± 0.3 that extends from 2 to 50 keV, with an unabsorbed luminosity of 1.3 × 1033 erg s-1 (d/8 kpc)2 in the 2-8 keV band. Despite possessing a cometary X-ray morphology that is typical of a pulsar wind nebula (PWN) in high-resolution Chandra imaging, our spatially resolved Chandra spectral analysis found no significant spectral softening along the filament as would be expected from particle synchrotron cooling. Coincident radio emission is detected using the Very Large Array at 5.5 and 8.3 GHz. We examine and subsequently discard a PWN or magnetic flux tube as the origin of G359.97-0.038. We use broadband spectral characteristics and a morphological analysis to show that G359.97-0.038 is likely an interaction site between the shell of Sgr A East and an adjacent molecular cloud. This is supported by CS molecular line spectroscopy and the presence of an OH maser.