The computation of Gaussian orthant probabilities has been extensively studied for low dimensional vectors. Here we focus on the high dimensional case and we present a two step procedure relying on both deterministic and stochastic techniques. The proposed estimator relies indeed on splitting the probability into a low dimensional term and a remainder. While the low dimensional probability can be estimated by fast and accurate quadrature, the remainder requires Monte Carlo sampling. We further refine the estimation by using a novel asymmetric nested Monte Carlo (anMC) algorithm for the remainder and we highlight cases where this approximation brings substantial efficiency gains. The proposed methods are compared against state-of-the-art methods in a numerical study, where we also analyse advantages and drawbacks of anMC. Finally the proposed method is applied to derive conservative estimates of excursion sets of expensive to evaluate deterministic functions under a Gaussian random field prior, without requiring a Markov assumption.