When it gets warmer, and the grass has begun growing again, simply placing a sheet of plastic over the area and weighting it down, will kill the grass extremely quickly. Here in the south, as early as April/May, this can happen in a week's time. Be sure your grass has greened up, the temperature should be at least 80 F for maximum effectiveness, and spray that area well with water before covering it. It really doesn't matter whether the plastic is clear or black. In fact, the clear is probably better, allowing the sunlight to penetrate and then creating the heat on the grass. The "steam" generated plus the lack of oxygen is what kills the grass, not the lack of light.
After the "heat-treatment" I would rake up what dead grass you can and then cover that area with cardboard or newspaper to retard seed/root rhizome growth.
