In a fuel-fired water heater with a standing pilot burner and a motorized flue damper, a specially designed controller is utilized to prevent overheating of water stored in the tank portion of the water heater caused by the hot combustion gases continuously generated by the pilot burner during standby periods of the water heater in which its main fuel burner is not being fired. The controller has a selectively variable water temperature control set point temperature and is operable to sense both ambient temperature and the tank water temperature and to open the flue damper and/or keep it open, after the main burner is off, in response to the presence for a predetermined continuous time period of a predetermined relationship of at least the selected temperature control set point temperature and the sensed ambient temperature.