Apparatus and method for aspirating liquid from a container, e.g., a test tube or other vessel, includes apparatus for sensing that an aspiration probe tip is contacting the inside bottom wall of the container, whereby a maximum volume of liquid can be aspirated from the container. According to the invention, an encoder is mounted on the driven member of a motor (e.g., on a rotatably-driven drive shaft or a nut of a stepper or D.C. motor, or on a linearly-driven shaft of a linear actuator) used to rectilinearly advance the probe tip into the container. Such encoder produces a series of pulses indicating the speed and/or linear position of the probe tip. A motor controller responds to a predetermined change in pattern of pulses from the encoder indicating that movement of the probe tip has been resisted by the vessel bottom, whereby further movement of the driven member and of the probe connected thereto is arrested, and the probe tip remains in contact with the vessel bottom.