Look underneath your couch to see how much fabric is available to use as a patch. There can be as much as 4 inches extra where the fabric is stapled into place. Use scissors to cut a piece of this excess fabric with which to patch the burn hole.
Trim the damaged edges from the burn area with scissors. Trim away only what is necessary so the hole does not get much larger.
Cut the fabric to be used as a patch so it is slightly larger than the burned area. Slip this under the fabric surrounding the hole. Line up the patch so it matches the surrounding fabric. For instance, if your couch has textured lines running along the fabric, match up the lines. This way the patch will be less obvious.
Glue the patch into place with fabric glue. Run a bead of glue just under the edge of the hole and on top of the patch. Gently press it into place and wipe away any excess glue. Put a piece of plastic wrap on top of the patched area and place a phone book on top of that. Let it dry.
Remove the phone book and carefully take off the plastic wrap. The couch is now ready for use again.