Context. Massive colliding-wind binaries (CWBs) can be non-thermal sources. The emission produced in their wind-collision region (WCR) encodes information of both the shocks properties and the relativistic electrons accelerated in them. The recently discovered system Apep, a unique massive system hosting two Wolf-Rayet stars, is the most powerful synchrotron radio emitter among the known CWBs, being an exciting candidate to investigate the non-thermal processes associated with stellar wind shocks. Aims. We intend to break the degeneracy between the relativistic particle population and the magnetic field strength in the WCR of Apep by probing its hard X-ray spectrum, where inverse-Compton (IC) emission is expected to dominate. Methods. We observe Apep with NuSTAR for 60 ks and combine this with a re-analysis of a deep archival XMM-Newton observation to better constrain the X-ray spectrum. We use a non-thermal emission model to derive physical parameters from the results. Results. We detect hard X-ray emission consistent with a power-law component from Apep. This is compatible with IC emission produced in the WCR for a magnetic field of ≈ 105-190 mG, corresponding to a magnetic-to-thermal pressure ratio in the shocks of ≈ 0.007-0.021, and a fraction of ∼ 1.5 × 10 −4 of the total wind kinetic power being transferred to relativistic electrons. Conclusions. This is the first time that the non-thermal emission from a CWB is detected both in radio and high energies. This allows us to derive the most robust constraints of the particle acceleration efficiency and magnetic field intensity in a CWB so far, reducing the typical uncertainty of a few orders of magnitude to just within a factor of a few. This constitutes an important step forward in our characterisation of the physical properties of CWBs.