Method of manufacturing polymer-coated heat-sealable packaging material and sealed packages (e.g., carton, box, or bag packages for dry and liquid foods). The material includes a fiber base of packaging paper or packaging board, an outer polymeric heat sealing layer such as LDPE, and an inner polymeric water vapor barrier layer which is partly or totally formed of an amorphous cycloolefin copolymer. In the method, the cycloolefin copolymer water vapor barrier and the heat sealing layers are brought onto the fiber base by an extrusion step, and in accordance with the manufacturing method, they can be placed on one side of the fiber base or on both sides of the fiber base for achieving a moisture barrier on both sides of the package. Also, polymeric oxygen barrier layers, such as EVOH or polyamide layers, can be incorporated into the packaging material. By using the cycloolefin copolymer as an extruded water vapor barrier layer, curling of the packaging material is prevented.