Cost-sensitive classification is critical in applications where misclassification errors widely vary in cost. However, overparameterization poses fundamental challenges to the cost-sensitive modeling of deep neural networks (DNNs). The ability of a DNN to fully interpolate a training data set can render a DNN, evaluated purely on the training set, ineffective in distinguishing a cost-sensitive solution from its overall accuracy maximization counterpart. This necessitates rethinking cost-sensitive classification in DNNs. To address this challenge, this paper proposes a cost-sensitive adversarial data augmentation (CSADA) framework to make overparameterized models cost sensitive. The overarching idea is to generate targeted adversarial examples that push the decision boundary in cost-aware directions. These targeted adversarial samples are generated by maximizing the probability of critical misclassifications and used to train a model with more conservative decisions on costly pairs. Experiments on well-known data sets and a pharmacy medication image (PMI) data set, made publicly available, show that our method can effectively minimize the overall cost and reduce critical errors while achieving comparable overall accuracy. Funding: Research reported in this publication was supported by the National Library of medicine of the National Institutes of Health in the United States under award number R01LM013624. Data Ethics & Reproducibility Note: This paper abides by data ethics requirements. Data used are publicly available online at https://deepblue.lib.umich.edu/data/concern/data_sets/6d56zw997 . Codes to replicate the results of this paper are available at https://doi.org/10.24433/CO.2139841.v1 .