We have observed photon-assisted Cooper-pair tunneling in an atomic-scale Josephson junction formed between a superconducting Nb tip and a superconducting Nb sample in a scanning tunneling microscope (STM) at 30 mK. High-resolution tunneling spectroscopy data show a zero-bias conduction peak and other sharp sub-gap peaks from coupling of the STM junction to resonances in the electromagnetic environment. The sub-gap peaks respond to incident microwave radiation by splitting into multiple peaks with the position and height depending on the frequency and amplitude of the microwaves. The inter-peak spacing shows that the charge carriers are Cooper pairs, rather than quasiparticles, and the power dependence reveals that the current originates from photon-assisted phase-incoherent tunneling of pairs, rather than the more conventional phase-coherent tunneling of pairs that yields Shapiro steps.