When constructing a Wilson chain to represent a quantum impurity model, the effects of truncated bath modes are neglected. We show that their influence can be kept track of systematically by constructing an "open Wilson chain" in which each site is coupled to a separate effective bath of its own. As a first application, we use the method to cure the so-called mass-flow problem that can arise when using standard Wilson chains to treat impurity models with asymmetric bath spectral functions at finite temperature. We demonstrate this for the strongly sub-Ohmic spin-boson model at quantum criticality where we directly observe the flow towards a Gaussian critical fixed point.