Summary: An approximate semi-analytical technique is developed for determining the response first-passage time probability density function of a class of lightly damped nonlinear oscillators subject to evolutionary stochastic excitation. Specifically, relying on a Markovian approximation of the response energy envelope, and on a stochastic averaging treatment, yields a backward Kolmogorov equation governing the evolution in time of the oscillator reliability function. Next, the backward Kolmogorov equation is solved by employing an appropriate orthogonal basis in conjunction with a Galerkin projection scheme. It is noted that the technique can account for arbitrary evolutionary excitation forms, even of the non-separable type. The special case of an undamped oscillator, for which relevant analytical results exist in the literature, is also included and studied in detail. Further, Markovian approximation of the potential energy envelope is considered as well. In comparison with the conventional amplitude-based energy envelope formulation, the intermediate step of linearizing the nonlinear stiffness element is circumvented, thus reducing the overall approximation degree of the technique. An additional significant advantage of the potential energy envelope formulation relates to the fact that its degree of accuracy appears rather insensitive to the nonlinearity magnitude (at least in the considered examples). Pertinent Monte Carlo simulation data are included in the numerical examples as well for assessing the accuracy of the technique.