The deployment of diverse energy storage technologies, with the combination of daily, weekly and seasonal storage dynamics, allows for the reduction of carbon dioxide (CO2) emissions per unit energy provided. In particular, the production, storage and re-utilization of hydrogen starting from renewable energy has proven to be one of the most promising solutions for offsetting seasonal mismatch between energy generation and consumption. A realistic possibility for large-scale hydrogen storage, suitable for long-term storage dynamics, is presented by salt caverns. In this contribution, we provide a framework for modeling underground hydrogen storage, with a focus on salt caverns, and we evaluate its potential for reducing the CO2 emissions within an integrated energy systems context. To this end, we develop a first-principle model, which accounts for the transport phenomena within the rock and describes the dynamics of the stored energy when injecting and withdrawing hydrogen. Then, we derive a linear reduced order model that can be used for mixed-integer linear program optimization while retaining an accurate description of the storage dynamics under a variety of operating conditions. Using this new framework, we determine the minimum-emissions design and operation of a multi-energy system with H2 storage. Ultimately, we assess the potential of hydrogen storage for reducing CO2 emissions when different capacities for renewable energy production and energy storage are available, mapping emissions regions on a plane defined by storage capacity and renewable generation. We extend the analysis for solar- and wind-based energy generation and for different energy demands, representing typical profiles of electrical and thermal demands, and different CO2 emissions associated with the electric grid.