As a chemist I can tell you that we do liquid-liquid extractions for the purification of organic molecules regularly. For the case you mention, you have a biphasic (meaning there are two distinct liquid layers that do not mix) mixture in a reaction tube. The desired product is contained in the bottom layer. The easiest way to isolate the bottom layer would be to use a glass or plasic pipet to draw out the bottom layer and put it in a separate tube or vial. Another possible way to separate then would be to decant the top layer which means to carefully pour off the top layer thus leaving the bottom layer only in the tube.