The focus of this study is on estimating the multivariate extreme value distributions associated with a vector of mutually correlated non-stationary Gaussian processes. This involves computing the joint crossing statistics of the vector processes by assuming the crossings to be Poisson counting processes. A mathematical artifice is adopted to take into account the dependencies that exist between the crossings of the processes. The crux in the formulation lies in the evaluation of a four-dimensional integral, which can be computationally expensive. This difficulty is bypassed by using saddlepoint approximation to reduce the dimension of the integral to be numerically computed to just two. The developments are illustrated through a numerical example and are validated using Monte Carlo simulations. © 2016 The Authors.