Estimating the currents that underlie the field potentials captured by electroencephalography (EEG), i.e., EEG source localization, is an ill-conditioned inverse problem. Existing solutions consider spatial continuity constraints, dynamic modeling, or sparsity constraints. The computational cost of combining these approaches, however, poses a challenge for practical applications. We propose a new computationally efficient EEG source localization method that employs spatial covariance estimation, state-space modeling, and sparsity-enforcing priors. We validate the performance of our method using both simulated and experimentally recorded EEG data. Our approach provides substantial performance improvements over existing methods and thereby facilitates practical applications in both neuroscience and medicine.