Present attack methods can make state-of-the-art classification systems based on deep neural networks misclassify every adversarially modified test example. The design of general defense strategies against a wide range of such attacks still remains a challenging problem. In this paper, we draw inspiration from the fields of cybersecurity and multi-agent systems and propose to leverage the concept of Moving Target Defense (MTD) in designing a meta-defense for 'boosting' the robustness of an ensemble of deep neural networks (DNNs) for visual classification tasks against such adversarial attacks. To classify an input image, a trained network is picked randomly from this set of networks by formulating the interaction between a Defender (who hosts the classification networks) and their (Legitimate and Malicious) users as a Bayesian Stackelberg Game (BSG). We empirically show that this approach, MTDeep, reduces misclassification on perturbed images in various datasets such as MNIST, FashionMNIST, and ImageNet while maintaining high classification accuracy on legitimate test images. We then demonstrate that our framework, being the first meta-defense technique, can be used in conjunction with any existing defense mechanism to provide more resilience against adversarial attacks that can be afforded by these defense mechanisms. Lastly, to quantify the increase in robustness of an ensemble-based classification system when we use MTDeep, we analyze the properties of a set of DNNs and introduce the concept of differential immunity that formalizes the notion of attack transferability.