The seasonal evolution of the Asian monsoon precipitation is crucial to Asian agriculture production and social economy. Based on the observations, we assess the performances of latest version of the IAP/LASG Flexible Global Ocean-Atmosphere (FGOALS-g3) compared with FGOALS-g2 in simulating the Asian monsoon seasonal evolution. Compared with a atmospheric experiment driven by historical monthly mean FGOALS-g3 SST, the influence of air-sea coupling on monsoon seasonal evolution is also studied. The results show that FGOALS-g3 has improved the ability to simulate the precipitation annual cycle in South Asia and the Northwest Pacific region which may be relevant to single atmospheric models, but does not improve in other regions. The biases of precipitation annual cycle and timing of monsoon from South Asian to Northwest Pacific and precipitation annual cycle in South East Asian are improved by air-sea coupling process. Compared with FGOALS-g2, the ability to simulate Northwest Pacific monsoon onset and withdrawal, Arabian Sea monsoon withdrawal and from Indian Peninsula to the South China Sea monsoon peak are significantly improved in FGOALS-g3. The delayed bias onset in South Asia and Indo-China Peninsula is increased in FGOALS-g3 relative to FGOALS-g2, which is associated with weak Somali jet caused by dry bias in African land at May. In the Northwest Pacific, the bias of monsoon onset simulation (delayed in the West and advanced in the East) is significantly reduced in FGOALS-g3, mainly because the SST bias in tropical Pacific is improved, and the local Hadley circulation is strengthened, resulting in anomalous subsidence in the region, which reduces the wet bias in January on the west side, the relative precipitation increases, and the improvement of wet bias in the East is the result of compensation between the dry biases in May and January. The improvement of the tropical SST pattern in the coupled model has great implication to improve the simulation ability of the Asian monsoon precipitation annual cycle.