Perhaps a better solution is to use a wooden golf tee. Again, put glue in the hole and a little on the end of the tee, tap it into the hole and proceed as described above. I use wooden chopsticks from the restaurant as small dowels for a variety of purposes. If the screw is large enough, or you want to whittle it down you may be able to use a chopstick for this. I understand a modern version of this is to use pieces of plastic cable ties instead of toothpick.
Some people suggest packing steel wool in the hole.
I've never tried this, but if you're in a hurry you might want to try it. Lee Valley sells a tool named The Plugger for this. Its a conical drill bit with a taper the same as a standard pencil sharpener. You drill out the hole, then sharpen a piece of dowel in a pencil sharpener, put glue in the hole, insert the dowel and proceed as described above.