Deep Neural Networks, despite their great success in diverse domains, are provably sensitive to small perturbations on correctly classified examples and lead to erroneous predictions. Recently, it was proposed that this behavior can be combatted by optimizing the worst case loss function over all possible substitutions of training examples. However, this can be prone to weighing unlikely substitutions higher, limiting the accuracy gain. In this paper, we study adversarial robustness through randomized perturbations, which has two immediate advantages: (1) by ensuring that substitution likelihood is weighted by the proximity to the original word, we circumvent optimizing the worst case guarantees and achieve performance gains; and (2) the calibrated randomness imparts differentially-private model training, which additionally improves robustness against adversarial attacks on the model outputs. Our approach uses a novel density-based mechanism based on truncated Gumbel noise, which ensures training on substitutions of both rare and dense words in the vocabulary while maintaining semantic similarity for model robustness.