When analyzing data that has been perturbed for privacy reasons, one is often concerned about its usefulness. Recent research on differential privacy has shown that the accuracy of many data queries can be improved by post-processing the perturbed data to ensure consistency constraints that are known to hold for the original data. Most prior work converted this post-processing step into a least squares minimization problem with customized efficient solutions. While improving accuracy, this approach ignored the noise distribution in the perturbed data. In this paper, to further improve accuracy, we formulate this post-processing step as a constrained maximum likelihood estimation problem, which is equivalent to constrained L1 minimization. Instead of relying on slow linear program solvers, we present a faster generic recipe (based on ADMM) that is suitable for a wide variety of applications including differentially private contingency tables, histograms, and the matrix mechanism (linear queries). An added benefit of our formulation is that it can often take direct advantage of algorithmic tricks used by the prior work on least-squares post-processing. An extensive set of experiments on various datasets demonstrates that this approach significantly improve accuracy over prior work.