Scrape off any splinters that line the hole with a razor blade. Leave the inside of the hole rough so the putty has plenty of friction to stick to.
Pick up a silver dollar-sized amount of wood putty with a plastic knife. Pat it down into the hole, pushing the putty down into the hole as far as you can. Wipe off any excess putty with a damp sponge. Allow it to cure overnight. Repeat this entire process twice more.
Run a 220-grit sanding block over the hole. Wipe off the dust with a tack cloth.
Dip a 1/4-inch nylon brush into the paint or finish of your choice. Dab it over the hole. Allow it 12 hours to dry, and then repeat.