Nanotechnology, an innovative field focused on nanosized materials, is combined with plant biotechnology through the green chemical strategy of synthesizing plant-induced nanoparticles (NPs). Synthesizing these NPs through novel, cost-effective, and eco-friendly methods plays a significant role in degrading and eliminating organic pollutants. The presence of dangerous biological agents and chemicals in water that surpass the standard threshold and could potentially impact human health and the environment is called water pollution. In the current study, we synthesized copper–silver bimetallic nanoparticles (BMNPs) using a novel, robust, and inexpensive method with leaf broth of Azadirachta indica as both the reducing and capping agent. Scanning electron microscopy and transmission electron microscopy investigations were used to examine the morphology of the synthesized BMNPs, and results indicate that synthesized NPs are in spherical core–shell morphology with a size of 20 nm. Research using Fourier transform infrared spectroscopy indicates that the capping and stabilization of NPs are achieved by the presence of bioactive functional groups including phenolic, amines, and aromatic ring. The activities of synthesized Cu–Ag NPs in degradation of malachite green (MG) dye were tested by the advanced oxidation process using peroxomonosulfate (PMS) as an oxidant. Activation parameters were computed, and pseudo first-order kinetics were used to describe the degradation of MG in the BMNPs/PMS system. Under optimal reaction conditions, the BMNPs/PMS system achieved a maximum MG degradation efficiency of 81% within 50 minutes. The results of the work promise excellent environmental safety for water systems against dye pollution.