In plasmas, electronic states can be well-localized bound states or itinerant free states, or something in between. In self-consistent treatments of plasma electronic structure such as the average-atom model, all states must be accurately resolved in order to achieve a converged numerical solution. This is a challenging numerical and algorithmic problem in large part due to the continuum of free states which is relatively expensive and difficult to resolve accurately. Siegert states are an appealing alternative. They form a complete eigenbasis with a purely discrete spectrum while still being equivalent to a representation in terms of the usual bound states and free states. However, many of their properties are unintuitive, and it is not obvious that they are suitable for self-consistent plasma electronic structure calculations. Here it is demonstrated that Siegert states can be used to accurately solve an average-atom model and offer advantages over the traditional finite-difference approach, including a concrete physical picture of pressure ionization and continuum resonances.