Sustainable development requires not only the use of sustainable energy resources, but also the efficient use of all energy resources. The latter should be reached by considering the concept of energy as well as exergy - the true magnitude of thermodynamic losses. Exergy describes the quality of an energy flow as the percentage that can be completely transformed into any other form of energy. Reduction of exergy losses represents a more efficient use of energy resources, which is essential in the long run, but it is not captured by standard energy costs, which are crucial in the short run. In this paper, exergy analysis is used in the context of a multi-carrier energy district to match the supply and demand not only in quantity but also in quality. The innovative contribution of this paper is the offering of a trade-off between reducing exergy losses and energy costs, thereby attaining sustainability of the energy district. A mixed-integer programming problem considering several energy devices is formulated to minimize a weighted sum of exergy losses and energy costs while satisfying time-varying user demands. The problem is solved by branch-and-cut. Numerical results demonstrate that the optimized operation of the energy devices makes the energy district sustainable in terms of exergy efficiency and costs.