In this paper, a new Probabilistic Sensitivity Analysis (PSA) approach based on the concept of relative entropy is proposed for design under uncertainty. The relative entropy based method evaluates the impact of a random variable on a design performance by measuring the divergence between two probability density functions of the performance response, obtained before and after the variation reduction of the random variable. The method can be applied both over the whole distribution of a performance response [called global response probabilistic sensitivity analysis (GRPSA)] and in any interested partial range of a response distribution [called regional response probabilistic sensitivity analysis (RRPSA)]. Such flexibility of our approach facilitates its use under various scenarios of design under uncertainty, for instance in robust design, reliability-based design, and utility optimization. The proposed method is applicable to both the prior-design stage for variable screening when a design solution is yet identified and the post-design stage for uncertainty reduction after an optimal design has been determined. The saddlepoint approximation approach is introduced for improving the computational efficiency of applying our proposed method. The proposed method is illustrated and verified by numerical examples and industrial design cases.