Prepare the Soil
Prepare the damaged area by removing all dead grass and 1" to 2" of soil 3/4" past the edge of the dead grass. So you won't have to cut the sod in odd shapes, remove the dead grass so that the prepared area forms a square. Cut the edges of the area straight.
Water the area, then amend the soil by working in either compost, leaf mold or fine-milled peat moss to a depth of 3" to 5". Be sure to add enough to make up for the soil you removed, but still leave enough space so that your sod will be level with the rest of the yard. That will probably mean leveling the amendment to 1" lower than soil level (depending on the thickness of the soil in your sod).
Compact the amended area by tamping with a board or rolling it with a half -filled water roller.