Reliability analysis for engineering structures subjected to disastrous stochastic dynamical actions is of paramount importance for the performance-based decision-making of design, and has long been one of the major challenges in civil and various engineering fields. In the present paper, a novel method based on the globally-evolving-based generalized density evolution equation (GE-GDEE) is proposed to capture the time-variant first-passage reliability of high-dimensional nonlinear systems under non-white-noise excitations. The GE-GDEE has been established for the transient probability density function (PDF) of an arbitrary path-continuous process, e.g., one response of interest for a high-dimensional system, as a one- or two-dimensional partial differential equation. From this perspective, an absorbing-boundary process corresponding to the response of interest under a given safe domain can be constructed, and then its GE-GDEE can be developed. The equivalent drift coefficient in the GE-GDEE is a conditional expectation of the original drift function under the non-failure condition for the response of interest. It can be estimated by some feasible numerical approaches based on data from hundreds of representative deterministic dynamical analyses of the underlying high-dimensional system. Then, the GE-GDEE can be solved numerically to obtain the transient PDF of the absorbing boundary process and time-variant first-passage reliability further. A numerical example is illustrated to verify the efficiency and accuracy of the proposed method. It demonstrates remarkably the high accuracy of the failure probability even for rare events, which are achieved with only relatively small number of deterministic dynamic analyses for general high-dimensional nonlinear systems. Problems to be further studied are finally discussed.