We report new Chandra hard X-ray (>2 keV) and Jansky Very Large Array C-band observations of the nuclear superbubble of NGC 3079, an analog of the "Fermi bubble" in our Milky Way. We detect extended hard X-ray emission on the southwest (SW) side of the galactic nucleus with coherent multiwavelength features in radio, Hα, and soft X-ray. The hard X-ray feature has a cone shape with possibly a weak cap, forming a bubble-like structure with a diameter of ~1.1 kpc. A similar extended feature, however, is not detected on the northeast (NE) side, which is brighter in all other wavelengths such as radio, Hα, and soft X-ray. Scattered photons from the nuclear region or other nearby point-like X-ray bright sources, inverse Compton emission from cosmic-ray (CR) electrons via interaction with the cosmic microwave background, or any individually faint stellar X-ray source populations, cannot explain the extended hard X-ray emission on the SW side and the strongly NE/SW asymmetry. A synchrotron emission model, plus a thermal component accounting for the excess at ~1 keV, can well characterize the broadband radio/hard X-ray spectra. The broadband synchrotron spectra do not show any significant cutoff, and even possibly slightly flatten at higher energy. This rules out a loss-limited scenario in the acceleration of the CR electrons in or around this superbubble. As the first detection of kiloparsec-scale extended hard X-ray emission associated with a galactic nuclear superbubble, the spatial and spectral properties of the multiwavelength emissions indicate that the CR leptons responsible for the broadband synchrotron emission from the SW bubble must be accelerated in situ, instead of transported from the nuclear region of the galaxy.