An apparatus (1) for extracting fluids from a body tissue or organ having an aperture, including a seat (26) for positioning said tissue, and a pressure membranes (17,22) for applying pressure to said tissue while leaving said tissue aperture open for the exit of said fluids. The surface (23) of the supporting surface of the tissue is shaped as a funnel or cone, so that the apex is directed downward and the base (29) delimits the supporting surface of the tissue; the apex region being constituted by a hollow cylinder (22) in order to allow the passage of the umbilical cord of a placenta or of the vessel for the drainage of fluids from the placenta or organ or tissue; the supporting surface being provided so as to avoid producing obstructions or collapse of the surface vessels of the placenta or organ or tissue, so as to avoid hindering the flow of blood or fluids