Computational spectral imaging aims to reconstruct the entire 3D spectral cube from a few compressive measurements. Recently different spectral imaging modalities have been developed by exploiting the wavelength-dependent behavior of diffractive lenses. Another line of development in this area is provided by spectral filter arrays which has enabled multi-spectral sensors. In this work, we develop a new compressive spectral imaging modality by exploiting a diffractive lens with a multi-spectral sensor. To reconstruct the spectral cube from these compressive measurements, a model-based fast sparse recovery algorithm is also developed. The performance of the proposed technique is illustrated for the visible range using different spectral filter array configurations and number of measurements. The results demonstrate that significant performance improvement can be achieved over a recent imaging technique with diffractive lenses, while also enabling snapshot imaging with simpler and more compact designs.