Hyperspectral images (HSIs) can reflect the spectral characteristics of objects in multiple bands, which can be used in various tasks, including classification, material detection and identification, and geological exploration. However, due to hardware limitations, spatial data have commonly been partially discarded to obtain more spectral information. Therefore, the enhancement of spatial resolution is often contemplated through the application of super-resolution algorithms. In view of this, this study proposes a diffusion model-assisted multi-scale spectral attention network (DMSANet) to increase the HSI resolution in the spatial dimension while preserving spectral information as much as possible. For the first time, a diffusion model is combined with deep networks to solve the HSI super-resolution problem, which enhances the spatial texture details of the output image using a layer-by-layer super-resolution mechanism of Markov chains. In addition, a multi-scale attention block that can integrate multiple receptive fields to extract spectral features of HSIs is designed, which enhances spectral information details. Extensive evaluations and comparisons on three benchmark datasets demonstrate that the proposed DMSANet can achieve superior performance compared with the existing methods.