The purpose of this study was to develop a resource recovery procedure for recovering copper and zinc from dust produced by copper smelting furnaces during the manufacturing of copper-alloy wires. The concentrations of copper in copper-containing dust do not meet the regulation standards defined by the Taiwan Environmental Protection Administration; therefore, such waste is classified as hazardous. In this study, the percentages of zinc and copper in the dust samples were approximately 38.4% and 2.6%, respectively. To reduce environmental damage and recover metal resources for industrial reuse, acid leaching was used to recover metals from these inorganic wastes. In the first stage, 2 N of sulphuric acid was used to leach the dust, with pH values controlled at 2.0–3.0, and a solid-to-liquid ratio of 1:10. The results indicated that zinc extraction efficiency was higher than 95%. A selective acid leaching process was then used to recover the copper content of the residue after filtration. In the second stage, an additional 1 N of sulphuric acid was added to the suspension in the selective leaching process, and the pH value was controlled at 1.5–2.0. The reagent sodium hydroxide (2 N) was used as leachate at a pH greater than 7. A zinc hydroxide compound formed during the process and was recovered after drying. The yields for zinc and copper were 86.9–93.5% and 97.0–98.9%, respectively.