The polymer residue remaining on chemical-vapor-deposited graphene after its transfer to the substrate and subsequent lithographic patterning tends to cause problems such as decrease in electron mobility, and unwanted doping. In this study, by using a controllable low-energy Ar+ ion beam (9.5 eV), the residue was cleaned perfectly without damaging the graphene surface. Further, a back-gate graphene field-effect transistor fabricated on the Ar+-ion-cleaned graphene surface showed about 4 times higher drain current than that showed by a similar transistor fabricated on pristine graphene. We believe that the technique used in this study can be useful in preventing the problems caused by the residue remaining on the graphene surface and can be applied not only to the processing of next-generation graphene-based electronics but also to other 2D materials-based electronic material processing.