Wipe away any debris, such as paint, sawdust or finish.
Insert the nozzle of a tube of wood filling compound (such as epoxy) into the hole.
Squeeze the tube of wood filler until the hole is completely filled. Allow the filler to dry before continuing.
Sand the excess wood filler off the front of the drawer once it has dried. You may then apply a coat of paint or finish to match the rest of the drawer.
Find a plug-cutting attachment that is the same diameter as the drawer's hole. Insert the attachment into a drill.
Find a scrap piece of wood that closely matches the wood color of the drawer. If the drawer is painted, you can use any wood you have on hand.
Drill down into the piece of wood with the plug-cutting bit. Remove the plug of wood you just bored.
Dab a small amount of wood glue around the outside of the wood plug. Insert the plug into the drawer hole until it is flush with the front of the drawer. Allow it to dry.
Brush a small amount of paint or wood finish over the plug if you wish to make it match the rest of the drawer.
Smear a small amount of wood glue on the back edge of a wood button. Wood buttons are press-on tabs that are used to cover but not hide holes.
Push the wood button into the drawer hole.
Tap the button gently with a hammer to drive it all the way into place. Allow the glue to dry.