In this paper, a modified multi-physics method for transient analysis of high-power microwaves (HPM) gas breakdown is proposed. Distinguished from previous works, the proposed method couples the plasma fluid equations with Maxwell's equations to fully consider the interaction between plasma and electromagnetic waves. To perform the numerical simulation, the spectral-element time-domain method is employed, which has the advantages of spectral accuracy and block diagonal mass matrix. Numerical simulations are conducted to demonstrate the accuracy of the proposed method. Moreover, with an external DC magnetic field, HPM breakdown can be effectively delayed by increasing its breakdown threshold. Simultaneously, the phase shift of electromagnetic waves during the HPM breakdown can also be controlled by the external DC magnetic field, which can improve the quality of the phase-modulated signal in HPM illumination. This proposed framework is expected to provide an effective numerical tool for analyzing the microwave propagation characteristics and suppressing the HPM breakdown in gas-filled microwave devices.