We introduce an innovative approach that incorporates a Distributionally Robust Learning (DRL) approach into Cox regression to enhance the robustness and accuracy of survival predictions. By formulating a DRL framework with a Wasserstein distance-based ambiguity set, we develop a variant Cox model that is less sensitive to assumptions about the underlying data distribution and more resilient to model misspecification and data perturbations. By leveraging Wasserstein duality, we reformulate the original min-max DRL problem into a tractable regularized empirical risk minimization problem, which can be computed by exponential conic programming. We provide guarantees on the finite sample behavior of our DRL-Cox model. Moreover, through extensive simulations and real world case studies, we demonstrate that our regression model achieves superior performance in terms of prediction accuracy and robustness compared with traditional methods.