We observe 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 subgap peaks from coupling of the STM junction to resonances in the electromagnetic environment. The subgap 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 interpeak 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.