It is uncommon for a sink overflow drain to become clogged. However, if it does, a plunger will not work on most sinks because of the angle and area of the overflow drain hole. With simple tools you can unblock an overflow drain without damaging your sink.
Place the small end of the plastic funnel inside the overflow drain hole. This will prevent scratches on the sink from feeding and turning the auger. Push it in firmly so it stays in place.
Feed the end of a plumber's snake or wire auger through the funnel and into the drain hole. If the auger does not fit through the funnel, cut the end off of the funnel until it is wide enough for the auger to fit. If you are clearing a bathtub overflow drain, you will need to remove the plate covering the drain, and you won't need to use a funnel.
Feed the auger as far as you can into the drain. If the drain has any sharp curves in it, turn the auger while you push. The auger will stop or slow once you have reached the blockage.
Clear the drain by spinning the auger handle while attempting to push through the blockage. Once you are past the blockage, alternate pushing and pulling through the drain. This will break up the larger chunks of whatever is blocking the drain.
Pull out the auger, block the main sink drain, and fill the sink to the level of the overflow drain with hot water. Running water down the drain will clear the rest of the blockage. You can also use the funnel to pour cups of hot water in the hole. When the water flows freely through the drain, the blockage is clear.
For some drains, a straightened wire coat hanger can be used for unblocking if an auger is unavailable. A coat hanger is difficult to use if the drain is very coarse or has sharp curves.