The 1/3 subharmonic resonance response of Duffing oscillator with Coulomb dry friction under foundation excitation is investigated, and the approximate analytical solution of the subharmonic resonance of the system is obtained by using the incremental averaging method. Based on the approximate analytical solution of the primary resonance obtained by the averaging method, the approximate analytical solution of subharmonic resonance is solved by the averaging method according to the incremental equation, and the amplitude–frequency response equation of subharmonic resonance is obtained. It is found that the Coulomb friction affects the amplitude–frequency response of both the primary resonance and subharmonic resonance of the nonlinear dry friction system in the form of equivalent damping. The comparison between the approximate analytical solution and the numerical solution shows that the approximate analytical solutions of the primary resonance and subharmonic resonance are both in very good agreement with the numerical solution. The existence condition of the 1/3 subharmonic resonance for the nonlinear dry friction system is presented, and the stability of the steady-state solution of subharmonic resonance is also judged. Based on the approximate analytical solution, the effects of the nonlinear stiffness and the Coulomb friction on the amplitude–frequency response of resonance and critical frequency of 1/3 subharmonic resonance of the nonlinear dry friction system are analyzed in detail. The analysis results show that the incremental averaging method can effectively obtain the approximate analytical solution in unified form for the subharmonic resonance of nonlinear system with Coulomb friction.