Super-resolution flow reconstruction using state-of-the-art data-driven techniques is valuable for a variety of applications, such as subgrid/subfilter closure modeling, accelerating spatiotemporal forecasting, data compression, and serving as an upscaling tool for sparse experimental measurements. In the present work, a first-of-its-kind multiscale graph transformer approach is developed for mesh-based super-resolution (SR-GT) of reacting flows. The novel data-driven modeling paradigm leverages a graph-based flow-field representation compatible with complex geometries and non-uniform/unstructured grids. Further, the transformer backbone captures long-range dependencies between different parts of the low-resolution flow-field, identifies important features, and then generates the super-resolved flow-field that preserves those features at a higher resolution. The performance of SR-GT is demonstrated in the context of spectral-element-discretized meshes for a challenging test problem of 2D detonation propagation within a premixed hydrogen-air mixture exhibiting highly complex multiscale reacting flow behavior. The SR-GT framework utilizes a unique element + neighborhood graph representation for the coarse input, which is then tokenized before being processed by the transformer component to produce the fine output. It is demonstrated that SR-GT provides high super-resolution accuracy for reacting flow-field features and superior performance compared to traditional interpolation-based SR schemes.