In this study, copper oxide nanoparticles (CuO NPs), zinc oxide nanoparticles (ZnO NPs), and copper oxide/zinc oxide nanocomposites (CuO@ZnO NPs) were synthesized by green synthetic route where bioactive compounds inherently present in the leaf extract of Artemisia vulgaris act as stabilizing and reducing agents. Phytochemicals present in leaf extract were assessed by qualitative chemical tests and spectroscopic measurements. UV-visible spectroscopy, Fourier Transform Infrared (FTIR) spectroscopy, Energy Dispersive X-ray (EDX) spectroscopy, X-ray Diffraction (XRD), and Field Emission Scanning Electron Microscopy (FESEM) were used to characterize the as-synthesized nanomaterials, i.e., CuO, ZnO, and CuO@ZnO NPs. XRD pattern revealed the crystalline nature of nanoparticles. Based on the Debye–Scherrer formula, the sizes of CuO NPs, ZnO NPs, and CuO@ZnO NCs were found to be 17.24, 20.74, and 22.50 nm, respectively. The band gap of the as-prepared nanomaterials was measured using the Tauc plot. Using the nanomaterials, MB degradation was studied at room temperature under exposure to UV light. The degradation efficiency of CuO, ZnO, and 2% CuO@ZnO was found to be 52%, 68%, and 98%, respectively. Kinetic degradation process reveals that the CuO@ZnO NCs showed a better photocatalytic activity on MB dye with the degradation constant of 0.04124 min−1 compared to those of either constituent. Based on the findings, it was found that CuO@ZnO nanocomposites have the potential to degrade MB as an organic dye and can be used for wastewater treatment.