We just moved into a previously owned home. The previous owners did some bathroom upgrades and installed some nice vessel sinks and granite counters but today we discovered that when one of these vessel sinks is filled to the point that water goes into the little overflow drain, the water seems to just leak out from under the sink onto the counter top. In other words, it seems as if the overflow drain is not draining into a pipe that carries water away. It just leaks out through the crack between the bottom of the vessel sink and the counter. This can't be normal because what's the point of an overflow drain if not to help prevent water overflowing onto the counter and floor if the tap is left running too long?

Does anyone know what the problem is and what can we do to fix this? Any help would be greatly appreciated.

By the way, we did have a home inspection done before buying the house and I know that they checked all the sinks for leaking but obviously they did not try over-filling the sinks! I fear that all 6 vessel sinks that the previous homeowners installed are improperly installed and probably all leak the same way.

As you point out, there is a leak path in the drain-to-overflow connection. The drain assembly will need to be taken apart and reassembled properly again.

It looks like I'll have to remove the whole sink then. There's only one pipe under the sink so I'm guessing the problem is only going to be visible if I disconnect the pipe and remove the vessel sink. Then maybe I will see a problem when I look at the underside of the vessel sink.

it sounds like they used a vessel sink with an overflow but used a drain that does not have an overflow opening.
a sink with an overflow has an internal passage/s that lead to the portion of the drain that will accept the water. i don't want to panic you but if this is the case then you are in trouble as you can not use the correct drain if the bottom of the sink just butts to the countertop.