Joint feature selection and classification in an online setting is essential for time-sensitive decision making. However, most existing methods treat this coupled problem independently. Specifically, online feature selection methods can handle either streaming features or data instances offline to produce a fixed set of features for classification, while online classification methods classify incoming instances using full knowledge about the feature space. Nevertheless, all existing methods utilize a set of features, common for all data instances, for classification. Instead, we propose a framework to perform joint feature selection and classification on-the-fly, so as to minimize the number of features evaluated for every data instance and maximize classification accuracy. We derive the optimum solution of the associated optimization problem and analyze its structure. Two algorithms are proposed, ETANA and F-ETANA, which are based on the optimum solution and its properties. We evaluate the performance of the proposed algorithms on several public datasets, demonstrating (i) the dominance of the proposed algorithms over the state-of-the-art, and (ii) its applicability to broad range of application domains including clinical research and natural language processing.