H2S is a crucial industrial gas and also a neurotoxin which poses a significant threat to human life and safety in case of leakage. Therefore, it is essential to monitor the H2S concentration in real time during industrial production and daily life. In this study, we utilized a method that combined electron beam evaporation and annealing to deposit a CuO film on a two-port quartz surface acoustic wave (SAW) resonator, which was connected to peripheral circuits to build a SAW sensor. The sensor exhibited excellent sensitivity, selectivity, and stability, and had a response of −30 kHz to 4 ppm H2S at RH = 60% and room temperature. The superior performance is attributed to the strong chemical adsorption of CuO toward H2S. CuO reacts with H2S to form CuS, leading to an increase in the mass of the film, which causes the negative frequency response of the sensor. It is also revealed that environmental humidity has a significant impact on the sensor response, with higher humidity resulting in a stronger response.