We speed up the Monte Carlo simulation of static graph reliability models by adding graph reductions to zero-variance importance sampling (ZVIS) approximation techniques. ZVIS approximation samples the status of links sequentially, and at each step we check if series-parallel reductions can be performed. We present two variants of the algorithm and describe their respective advantages. We show that the method satisfies robustness properties as the reliability of links increases. We illustrate theoretically on small examples and numerically on large ones the gains that can be obtained, both in terms of variance and computational time.