Composites of copper (hydr)oxychlorides with graphite oxide or graphene were synthesized and used as adsorbents of hydrogen sulfide at dynamic conditions at ambient temperatures. The materials were extensively characterized before and after adsorption in order to link their performance to the surface features. X-ray diffraction, FTIR, thermal analysis, TEM, SEM/EDX, and adsorption of nitrogen were used. It was found that the composite with graphene has the most favorable surface features enhancing reactive adsorption of hydrogen sulfide. The presence of moisture in the H2S stream has a positive effect on the removal process owing to the dissociation process. H2S is retained on the surface via a direct replacement of OH groups and via acid-base reactions with the copper (hydr)oxide. Highly dispersed reduced copper species on the surface of the composite with graphene enhance activation of oxygen and cause formation of sulfites and sulfates. Higher conductivity of the graphene phase than that of graphite oxide helps in electron transfer in redox reactions.