We present an end-to-end Spatial-Temporal Graph Attention Network (STGAT) for non-invasive detection and width estimation of Cortical Spreading Depressions (CSDs) on scalp electroencephalography (EEG). Our algorithm, that we refer to as CSD Spatial-temporal graph attention network or CSD-STGAT, is trained and tested on simulated CSDs with varying width and speed ranges. Using high-density EEG, CSD-STGAT achieves less than 10.96% normalized width estimation error for narrow CSDs, with an average normalized error of 6.35%±3.08% across all widths, enabling non-invasive and automated estimation of the width of CSDs for the first time. In addition, CSD-STGAT learns the temporal and spatial features of CSDs simultaneously, which improves the "spatio-temporal tracking accuracy" (i.e., the defined detection performance metric at each electrode) of the narrow CSDs by up to 14%, compared to the state-of-the-art CSD-SpArC algorithm, with only one-tenth of the network size. CSD-STGAT achieves the best spatio-temporal tracking accuracy of 86.27%±0.53% for wide CSDs using high-density EEG, which is comparable to the performance of CSD-SpArC with less than 0.38% performance reduction. We further stitch the detections across all electrodes and over time to evaluate the "temporal accuracy". Our algorithm achieves less than 0.7% false positive rate in the simulated dataset with inter-CSD intervals ranging from 5 to 60 minutes. The lightweight architecture of CSD-STGAT paves the way towards real-time detection and parameter estimation of these waves in the brain, with significant clinical impact.