Centaurus A (Cen A) is the nearest radio galaxy discovered as a very-high-energy (VHE; 100 GeV–100 TeV) γ-ray source by the High Energy Stereoscopic System (H.E.S.S.). It is a faint VHE γ-ray emitter, though its VHE flux exceeds both the extrapolation from early Fermi-LAT observations as well as expectations from a (misaligned) single-zone synchrotron-self Compton (SSC) description. The latter satisfactorily reproduces the emission from Cen A at lower energies up to a few GeV. New observations with H.E.S.S., comparable in exposure time to those previously reported, were performed and eight years of Fermi-LAT data were accumulated to clarify the spectral characteristics of the γ-ray emission from the core of Cen A. The results allow us for the first time to achieve the goal of constructing a representative, contemporaneous γ-ray core spectrum of Cen A over almost five orders of magnitude in energy. Advanced analysis methods, including the template fitting method, allow detection in the VHE range of the core with a statistical significance of 12σ on the basis of 213 hours of total exposure time. The spectrum in the energy range of 250 GeV–6 TeV is compatible with a power-law function with a photon index Γ = 2.52 ± 0.13_stat ± 0.20_sys. An updated Fermi-LAT analysis provides evidence for spectral hardening by ΔΓ ≃ 0.4 ± 0.1 at γ-ray energies above 2.8^+1.0_−0.6 GeV at a level of 4.0σ. The fact that the spectrum hardens at GeV energies and extends into the VHE regime disfavour a single-zone SSC interpretation for the overall spectral energy distribution (SED) of the core and is suggestive of a new γ-ray emitting component connecting the high-energy emission above the break energy to the one observed at VHE energies. The absence of significant variability at both GeV and TeV energies does not yet allow disentanglement of the physical nature of this component, though a jet-related origin is possible and a simple two-zone SED model fit is provided to this end.