Electromagnetic source imaging (ESI) requires solving a highly ill-posed inverse problem. To seek a unique solution, traditional ESI methods impose various forms of priors that may not accurately reflect the actual source properties, which may hinder their broad applications. To overcome this limitation, in this article, a novel data-synthesized spatiotemporally convolutional encoder-decoder network (DST-CedNet) method is proposed for ESI. The DST-CedNet recasts ESI as a machine learning problem, where discriminative learning and latent-space representations are integrated in a CedNet to learn a robust mapping from the measured electroencephalography/magnetoencephalography (E/MEG) signals to the brain activity. In particular, by incorporating prior knowledge regarding dynamical brain activities, a novel data synthesis strategy is devised to generate large-scale samples for effectively training CedNet. This stands in contrast to traditional ESI methods where the prior information is often enforced via constraints primarily aimed for mathematical convenience. Extensive numerical experiments as well as analysis of a real MEG and epilepsy EEG dataset demonstrate that the DST-CedNet outperforms several state-of-the-art ESI methods in robustly estimating source signals under a variety of source configurations.