This is an era of data deluge with individuals and pervasive sensors acquiring large and ever-increasing amounts of data. Nevertheless, given the inherent redundancy, the costs related to data acquisition, transmission, and storage can be reduced if the per-datum importance is properly exploited. In this context, the present paper investigates sparse linear regression with censored data that appears naturally under diverse data collection setups. A practical censoring rule is proposed here for data reduction purposes. A sparsity-aware censored maximum-likelihood estimator is also developed, which fits well to big data applications. Building on recent advances in online convex optimization, a novel algorithm is finally proposed to enable real-time processing. The online algorithm applies even to the general censoring setup, while its simple closed-form updates enjoy provable convergence. Numerical simulations corroborate its effectiveness in estimating sparse signals from only a subset of exact observations, thus reducing the processing cost in big data applications.