Cut out the damaged sod with a masonry trowel. Lift up the damaged sod and place it in a wheelbarrow.
Till the bare area or break it up with a gardening fork, so the soil isn't compacted. Smooth over the area with your hand or the back of a rake.
Mix up in a bucket your preferred grass seed and compost. Spread a thin layer of seed/compost mixture over the bare area.
Water the bare area. Keep the grass seed area moist to prevent seeds from drying out. Dried up grass seeds will not germinate.
Use a masonry trowel to cut through the damaged sod. Lay the piece on a new piece of sod. Cut through the new piece of sod using the damaged piece as an outline. You can also place the sod piece in the bare spot and try to cut the piece so that it butts up against the surrounding sod.
Spread starter fertilizer over the bare area. Use a starter fertilizer that is high in phosphorous for strong root development. Work the starter fertilizer into the first 2 to 4 inches of soil. Smooth out the top with a rake or your hand.
Water the bare area and piece of sod, so that it is moist. Lay the sod piece on the bare area. Pat the sod down with your hand. If you are repairing a larger area, push a lawn roller over the area once the sod is dry.
Water the sod area until it is moist. Keep the sod moist for two weeks.