A hydrogen sulfide gas sensor based on Cu/Graphene composite membrane coated waist-enlarged tapered Photonic Crystal Fiber (PCF) is proposed and fabricated. The Mach-Zehnder Interferometer (MZI) is formed by fusion splicing a section of PCF which is sandwiched between two Single-Mode Fibers (SMF), and the air holes of PCF in the splicing regions are fully collapsed and formed two waist-enlarged tapers. The outside surface of PCF is coated with a layer of nano-Cu-deposited graphene by using a dip-coating technique. Cu-deposited graphene-coated PCF is to make the sensor produce high sensitivity. With the increasing concentration of hydrogen sulfide, the output wavelengths appear blue shift. In addition, a high hydrogen sulfide gas sensitivity of 8.5 pm/ppm, the limit of detection of 3.85 ppm and good linear relationship and selectivity are obtained within a measurement range of 0~80 ppm for H2S gas. The response and recovery time of the sensor are 92 s and 119 s, respectively. The experimental results show that the sensor has the advantages of low cost, high sensitivity and simple structure, especially suitable for low concentration and high sensitivity detection of hydrogen sulfide gas.