Electroencephalography (EEG) and magnetoencephalography noninvasively record scalp electromagnetic fields generated by cerebral currents, revealing millisecond-level brain dynamics useful for neuroscience and clinical applications. Estimating the currents that generate these fields, i.e., source localization, is an ill-conditioned inverse problem. Solutions to this problem have focused on spatial continuity constraints, dynamic modeling, or sparsity constraints. The combination of these key ideas could offer significant performance improvements, but substantial computational costs pose a challenge for practical application of such approaches. Here, we propose a new method for EEG source localization that combines: 1) covariance estimation for both source and measurement noises; 2) linear state-space dynamics; and 3) sparsity constraints, using 4) novel computationally efficient estimation algorithms. For source covariance estimation, we use a locally smooth basis alongside sparsity enforcing priors. For EEG measurement noise covariance estimation, we use an inverse Wishart prior density. We estimate these model parameters using an expectation–maximization algorithm that employs steady-state filtering and smoothing to expedite computations. We characterize the performance of our method by analyzing simulated data and experimental recordings of eyes-closed alpha oscillations. Our sparsity enforcing priors significantly improve estimation of both the spatial distribution and time course of simulated data, while improving computational time by more than 12-fold over previous dynamic methods. Our approach provides substantial performance improvements over existing methods using computationally efficient algorithms that will facilitate practical applications in both neuroscience and medicine.