An iterative method for estimating the failure probability for certain time-variant reliability problems has been developed. In the paper, the focus is on the displacement response of a linear oscillator driven by white noise. Failure is then assumed to occur when the displacement response exceeds a critical threshold. The iteration procedure is a two-step method. On the first iteration, a simple control function promoting failure is constructed using the design point weighting principle. After time discretization, two points are chosen to construct a compound deterministic control function. It is based on the time point when the first maximum of the homogenous solution has occurred and on the point at the end of the considered time interval. An importance sampling technique is used in order to estimate the failure probability functional on a set of initial values of state space variables and time. On the second iteration, the concept of optimal control function can be implemented to construct a Markov control which allows much better accuracy in the failure probability estimate than the simple control function. On both iterations, the concept of changing the probability measure by the Girsanov transformation is utilized. As a result the CPU time is substantially reduced compared with the crude Monte Carlo procedure.