A cassette and controller for accurately controlling fluid flow through an administration set while maintaining a closed fluid system. Inlet and outlet conduits are defined in a body of the cassette and inlet and outlet valves control fluid flow through the respective conduits. A rigid internal cavity of predetermined volume is also defined in the cassette body with an impermeable and flexible diaphragm spanning the cavity. A control conduit communicates with the internal cavity on an opposite side of the diaphragm from the fluid to alternately exert negative and positive pressure on the diaphragm. Negative pressure causes the diaphragm to conform to the shape of the internal cavity when the inlet valve is open to fill the cavity-shaped diaphragm with fluid from the inlet conduit. Positive pressure against the diaphragm expels fluid from the cavity through the outlet conduit when the outlet valve is opened. A controller for the cassette provides the positive and negative pressures and controls the inlet and outlet valves. Related methods of metering the fluid through the cassette are also disclosed.