The binary system $\eta$ Carinae is a unique laboratory in which to study particle acceleration to high energies under a wide range of conditions, including extremely high densities around periastron. To date, no consensus has emerged as to the origin of the GeV $\gamma$-ray emission in this important system. With a re-analysis of the full Fermi-LAT dataset for $\eta$ Carinae we show that the spectrum is consistent with a pion decay origin. A single population leptonic model connecting the X-ray to $\gamma$-ray emission can be ruled out. Here, we revisit the physical model of Ohm et al. (2015), based on two acceleration zones associated to the termination shocks in the winds of both stars. We conclude that inverse-Compton emission from in-situ accelerated electrons dominates the hard X-ray emission detected with NuSTAR at all phases away from periastron, and pion-decay from shock accelerated protons is the source of the $\gamma$-ray emission. Very close to periastron there is a pronounced dip in the hard X-ray emission, concomitant with the repeated disappearance of the thermal X-ray emission, which we interpret as being due to the suppression of significant electron acceleration in the system. Within our model, the residual emission seen by NuSTAR at this phase can be accounted for with secondary electrons produced in interactions of accelerated protons, in agreement with the variation in pion-decay $\gamma$-ray emission. Future observations with H.E.S.S., CTA and NuSTAR should confirm or refute this scenario.