We report the first detection of high-energy X-ray (E > 10 keV) emission from the Galactic center non-thermal filament G359.89–0.08 (Sgr A–E) using data acquired with the Nuclear Spectroscopic Telescope Array (NuSTAR). The bright filament was detected up to ∼50 keV during a NuSTAR Galactic center monitoring campaign. The featureless power-law spectrum with a photon index Γ ≈ 2.3 confirms a non-thermal emission mechanism. The observed flux in the 3-79 keV band is FX = (2.0 ± 0.1) × 10–12 erg cm–2 s–1, corresponding to an unabsorbed X-ray luminosity LX = (2.6 ± 0.8) × 1034 erg s–1 assuming a distance of 8.0 kpc. Based on theoretical predictions and observations, we conclude that Sgr A–E is unlikely to be a pulsar wind nebula (PWN) or supernova remnant-molecular cloud (SNR-MC) interaction, as previously hypothesized. Instead, the emission could be due to a magnetic flux tube which traps TeV electrons. We propose two possible TeV electron sources: old PWNe (up to ∼100 kyr) with low surface brightness and radii up to ∼30 pc or MCs illuminated by cosmic rays (CRs) from CR accelerators such as SNRs or Sgr A*.