Lift the splintered piece of wood from the surface of the furniture with the blade of a utility knife just enough so that you can apply glue underneath it.
Squeeze wood glue into the area beneath the splintered piece of wood and remove the utility knife blade.
Press the splintered piece down into place and clamp it with a wood clamp. Use a damp cloth to wipe off the excess glue that oozes out. Allow the glue to dry 24 hours, then remove the clamp.
Sand the repaired area with a fine-grit sanding block until it is smooth. Always sand in the direction of the wood grain. Wipe off the sanding dust with a tack cloth.
Brush two even coats of wood finish onto the repaired area of the furniture with a paintbrush. Allow the finish to dry at least four hours between coats. Wait 24 hours after the second coat before using the furniture piece.