Non-thermal hard X-ray and high-energy (HE; 1 MeV < E < 100 GeV) gamma-ray emission in the direction of Eta Carina has been recently detected using the INTEGRAL, AGILE and Fermi satellites. So far this emission has been interpreted in the framework of particle acceleration in the colliding wind region between the two massive stars. However, the existence of a very fast moving blast wave which originates in the historical 1843 Great Eruption provides an alternative particle acceleration site in this system. Here we explore an alternate scenario and find that inverse Compton emission from electrons accelerated in the blast wave can naturally explain both the flux and spectral shape of the measured hard X-ray and HE gamma-ray emission. This scenario is further supported by the lack of significant variability in the INTEGRAL and Fermi measured fluxes.