In this work, we report the engineering of sub 30 nm nanoparticles of copper oxide (CuO) and Zinc oxide (ZnO) and their nanocomposite, using a green method, to degrade an organic dye (methyl orange) that is responsible for water pollution. The plant extract of Eucalyptus globulus was used as a capping and stabilizing agent, as it is non-toxic, easy to use, and affordable. The percent purity and composition of the samples were found using Energy Dispersive Spectroscopy (EDS), which confirmed the formation of 75% CuO nanoparticles, 77.81% ZnO nanoparticles, and 77.34% ZnO/CuO nanocomposite in respective samples. Morphological analysis was achieved using Scanning Electron Microscopy (SEM). FTIR investigations revealed that the vibrations of CuO and ZnO nanoparticles and CuO/ZnO nanocomposite were observed at 568 cm−1, 617.9 cm−1, and 475 cm−1, respectively. The bandgap of the synthesized CuO and ZnO nanoparticles, and ZnO/CuO nanocomposite was analyzed using Diffuse Reflectance Spectroscopy (DRS), and found to be 3.36 eV, 1.83 eV, and 1.48 eV, respectively. Best photocatalytic activity confirmed that favorable conditions for the maximum degradation of methyl orange are at pH 12 and 0.02 g of the photocatalyst is required.