Decarbonization in buildings calls for advanced control strategies that coordinate on-site renewables, grid electricity, and thermal demand. Literature approaches typically rely on demand side management strategies or on active energy storage, like batteries. However, the first solution often neglects carbon-aware objectives, and could lead to grid overload issues, while batteries entail environmental, end-of-life, and cost concerns. To overcome these limitations, we propose an optimal, carbon-aware optimization strategy that exploits the building's thermal mass as a passive storage, avoiding dedicated batteries. Specifically, when a surplus of renewable energy is available, our strategy computes the optimal share of surplus to store by temporarily adjusting the indoor temperature setpoint within comfort bounds. Thus, by explicitly accounting for forecasts of building energy consumption, solar production, and time-varying grid carbon intensity, our strategy enables emissions-aware load shifting while maintaining comfort. We evaluate the approach by simulating three TRNSYS models of the same system with different thermal mass. In all cases, the results show consistent reductions in grid electricity consumption with respect to a baseline that does not leverage surplus renewable generation. These findings highlight the potential of thermal-mass-based control for building decarbonization.