A filling machine includes filling elements, each having a seal that seals a container's mouth, a filling valve, a gas valve, a liquid channel that, when the filling valve is opened, is in fluid communication with the container's interior, and a gas channel that, when the gas valve is open, is in fluid communication with the container's interior. The filling machine introduces an inert gas into the container's interior via the gas channel to create a preload pressure in the container such that the inert gas in the container dissolves in the liquid when the liquid is later introduced into the container. The filling machine is further configured to introduce a volume of liquid into the container's interior through the liquid channel at a filling pressure that exceeds the preload pressure.