This paper presents a new computationally scalable framework for accurate solution of chance constrained programs. A Bernstein approximation is used to transcribe the chance constraint into a deterministic constraint, relying heavily upon the evaluation of exponential moment generating functions. This computationally burdensome task is readily handled with Markov chain Monte Carlo integration. To address the conservatism of the MCMC/Bernstein approach, a new split-exponential moment generating function is proposed, thereby significantly improving the optimality of the obtained approximation. It is shown through illustrative examples that the new split-Bernstein approach provides near-optimal results to chance constrained programs.