Select a leather repair kit that is water based, not solvent based, which will cause leather to crack. The kit should include a repair compound and tinted top coat. When choosing a kit, consider the color and finish of your couch, such as matte or smooth, and select accordingly.
Clean and condition the snagged area to give yourself a good base to work from. If you are in a hurry, at least use a cotton swab dipped in rubbing alcohol to clean around the damaged section.
Adhere any parts of the leather that have lifted away from the back with a small amount of repair compound and allow it to dry. Trim any rough edges with scissors or nail clippers.
Apply a small amount of the repair compound over the snag and use a wet finger or blunt-edged spreading tool to remove excess compound and smooth the edges. Allow it to dry completely and repeat the process in layers until the seams from the snag are filled to your satisfaction. Always allow the compound to dry completely between coats.
Smooth out any visible edges from the repair compound by swiping an alcohol pad lightly across the repaired area a few times. Be careful not to saturate the area or wipe too hard as you are only smoothing out the surface.
Apply the tinted top coat as directed by the manufacturer. Light coats that are allowed to fully dry in between will be more durable and blend better than one thick coat.
Apply leather conditioner over the area when the final coat has dried. Hereafter, avoid any cleaners that include solvents.