By virtue of the Liouville equation and Novikov theorem, the approximate Fokker-Planck equation was derived, and the analytic expression of the stationary probability distribution (SPD) was obtained in an asymmetric bistable system driven by cross-correlated multiplicative white noise and additive white noise with periodic signal. Based on the computed results, effects of the following parameters, including intensity of noises and their correlation, the amplitude and frequency of the periodic signal, and the asymmetric parameter of system on the SPD were investigated. Numerical results show that: 1) The intensity of noises and their correlation, amplitude of periodic signal, and asymmetric parameter of the system can induce the SPD transition from bimodal to unimodal in structure, i.e., a non-equilibrium phase transition is generated. 2) There is no non-equilibrium phase transition as frequency of the periodic signal varies. 3) The structure of the SPD exhibits a symmetrical structure at x=0 in the case of r=0, while the symmetry of SPD at x=0 is destroyed if the asymmetric parameter of the system r≠0.