Reliable neutron-induced-reaction cross sections of unstable nuclei are essential for nuclear astrophysics and applications but their direct measurement is often impossible. The surrogate-reaction method is one of the most promising alternatives to access these cross sections. In this work, we successfully applied the surrogate-reaction method to infer for the first time both the neutron-induced fission and radiative capture cross sections of ^{239}Pu in a consistent manner from a single measurement. This was achieved by combining simultaneously measured fission and γ-emission probabilities for the ^{240}Pu(^{4}He,^{4}He^{'}) surrogate reaction with a calculation of the angular-momentum and parity distributions populated in this reaction. While other experiments measure the probabilities for some selected γ-ray transitions, we measure the γ-emission probability. This enlarges the applicability of the surrogate-reaction method.