We report on the first hard X-ray detection of the Geminga pulsar above 10 keV using a 150 ks observation with the Nuclear Spectroscopic Telescope Array (NuSTAR) observatory. The double-peaked pulse profile of non-thermal emission seen in the soft X-ray band persists at higher energies. Broadband phase-integrated spectra over the 0.2-20 keV band with NuSTAR and archival XMM-Newton data do not fit to a conventional two-component model of a blackbody plus power law, but instead exhibit spectral hardening above ~5 keV. We find that two spectral models fit the data well: (1) a blackbody (kT1 ~ 42 eV) with a broken power law (Γ1 ~ 2.0, Γ2 ~ 1.4 and Ebreak ~ 3.4 keV) and (2) two blackbody components (kT1 ~ 44 eV and kT2 ~ 195 eV) with a power-law component (Γ ~ 1.7). In both cases, the extrapolation of the Rayleigh-Jeans tail of the thermal component is consistent with the UV data, while the non-thermal component overpredicts the near-infrared data, requiring a spectral flattening at E ~ 0.05-0.5 keV. While strong phase variation of the power-law index is present below ~5 keV, our phase-resolved spectroscopy with NuSTAR indicates that another hard non-thermal component with Γ ~ 1.3 emerges above ~5 keV. The spectral hardening in non-thermal X-ray emission as well as spectral flattening between the optical and X-ray bands argue against the conjecture that a single power law may account for multi-wavelength non-thermal spectra of middle-aged pulsars.