Reliability-based mission planning aims to identify an optimal path for off-road autonomous ground vehicles (AGVs) under uncertain terrain environment, while satisfying specific mission mobility reliability (MMR) constraints. The evaluation of MMR during path planning poses computational challenges for practical applications. This paper presents an efficient reliability-based mission planning using an outcrossing approach that has the same computational complexity as deterministic mission planning. A Gaussian random field is employed to represent the spatially dependent uncertainty sources in the terrain environment. The latter are then used in conjunction with a vehicle mobility model to generate a stochastic mobility map. Based on the stochastic mobility map, outcrossing rate maps are generated using the outcrossing concept which is widely used in time-dependent reliability. Integration of the outcrossing rate map with a rapidly-exploring random tree (RRT*) algorithm, allows for efficient path planning of AGVs subject to MMR constraints. A reliable RRT* algorithm using the outcrossing approach (RRT*-OC) is developed to implement the proposed efficient reliability-based mission planning. Results of a case study verify the accuracy and efficiency of the proposed algorithm.