Periodic metallic structures are known to support resonant extraordinary optical transmission (EOT). When covered with graphene, these structures can be employed to effectively manipulate the light. In this paper, we propose an analytical circuit model for graphene-covered 1-D metallic gratings. By using the circuit theory, we demonstrate that 1-D periodic array of cut-through slits, which are covered by a continuous graphene sheet, exhibits tunable EOT resonance for TM polarization whose amplitude, unlike its spectral position, can be dynamically tuned by varying the Fermi level of graphene. In this fashion, it is shown that placing a perfect reflector at the bottom of the graphene-covered metallic grating results in the realization of a graphene-based absorber. By utilizing the circuit theory, it is illustrated that perfect absorption in the structure is not exclusive to the TM polarization, and it is shown that the TE polarized plane waves can be completely absorbed by duly adjusting the Fermi level of graphene. Criteria for the enhanced absorption are accordingly presented. Results of this paper may provide a useful tool for designing novel devices based on the graphene-covered metallic gratings, such as filters, modulators, and efficient absorbers.