Property variations in a structure strongly impact the macroscopic mechanical performance as regions with lower strength will be prone to damage initiation or acceleration. Consideration of the variability in material property is critical for high-resolution simulations of damage initiation and propagation. While the recent progressive damage analyses consider randomness in property fields, accurately quantifying the uncertainty in damage measures remains computationally expensive. Stochastic damage analyses require extensive sampling of random property fields and numerous replications of the underlying non-linear deterministic simulations. This paper demonstrates that a Quasi Monte Carlo (QMC) method, which uses a multi-dimensional low discrepancy Sobol sequence, is a computationally economical way to obtain the mean and standard deviations in cracks evolving in composites. An Extended Finite Element Method (XFEM) method with spatially random strength fields simulates the damage initiation and evolution in a model composite. We compared the number of simulations required for Monte Carlo (MC) and QMC techniques to measure the influence of input variability on the mean crack-length in an open-hole angle-ply tensile test. We conclude that the low discrepancy sampling and QMC technique converges substantially faster than traditional MC methods.