For China, the development of low-energy buildings is one of the necessary routes for achieving carbon neutrality. Combining photovoltaic (PV) with air source heat pump (ASHP) yields a great potential in providing heating and domestic hot water (DHW) supply in non-central heating areas. However, the diurnal and seasonal inconsistencies between solar availability and building heat load can severely affect the efficacy of solar energy systems. This study creates and numerically simulates a PV–ASHP system with thermal energy storage (TES) in transient system simulation software, TRNSYS. Experimental studies are conducted to validate the simulation model. The system’s yearly operational characteristics are simulated to reveal the energy conversion relationship between the system’s thermoelectric storage and heating and DHW demand. The results show that the synergy between heating and DHW simultaneously improves the direct utilization of solar energy compared to single heating. The yearly self-consumption and self-satisfaction rates of PV and the COP of the ASHP increase by 131.25%, 10.53% and 9.56%, respectively. Solar energy contributes 55.54% to the system, with a PV capacity of 82 W per square meter of building area. This study provides fresh approaches to developing flexible building-integrated PV–ASHP technologies and balance of the energy exchange among the PV, building load and TES.