Remove any ashes or residue from the rug with a hand vacuum.
Snip the burned fibers from the rug with sharp scissors. Hold each rug fiber straight up and cut only the burned area from the rug. Avoid cutting into the rug pad.
Clip small pieces of rug fibers from a corner or underneath side of the rug to place inside the affected area.
Mix 1 tsp. liquid dish soap into a quart of warm water. Wet a cloth with the soapy mixture. Dab the outside edge of the hole with the cloth to remove any black residue that surrounds the affected area. Continue blotting, removing the black stain.
Wet a clean cloth with water and pat it over the just clean area of carpet to remove soap residue. Pat the area dry with a towel.
Pour glue inside the hole in the carpet, covering it completely. Grasp one rug fiber with a pair of tweezers. Place one end of the fiber into the glue. Repeat the process with the remaining rug fiber, filling the hole completely. Avoid walking or sweeping the area for 24 hours to allow the glue time to bond to the fibers.