Some of the major home improvement stores sell patches for up to 8" x 8" areas. These patches consist of a perforated aluminum plate with fiberglass adhesive mesh tape over them. The fiberglass mesh tape extends over the aluminum piece about 4" all around. I paid $5 for one from Home Depot.
Simply remove the patch, apply it to the wall, and apply mud over it. Sand, remud over a larger area to blend the patch into the wall. I would recommend using the small abrasive sanding block (foam rubber with coarse finish).