In high-energy astronomy, spectro-imaging instruments such as X-ray detectors allow investigation of the spatial and spectral properties of extended sources including galaxy clusters, galaxies, diffuse interstellar medium, supernova remnants, and pulsar wind nebulae. In these sources, each physical component possesses a different spatial and spectral signature, but the components are entangled. Extracting the intrinsic spatial and spectral information of the individual components from this data is a challenging task. Current analysis methods do not fully exploit the 2D-1D (x, y, E) nature of the data, as spatial information is considered separately from spectral information. Here we investigate the application of a blind source separation (BSS) algorithm that jointly exploits the spectral and spatial signatures of each component in order to disentangle them. We explore the capabilities of a new BSS method (the general morphological component analysis; GMCA), initially developed to extract an image of the cosmic microwave background from Planck data, in an X-ray context. The performance of the GMCA on X-ray data is tested using Monte-Carlo simulations of supernova remnant toy models designed to represent typical science cases. We find that the GMCA is able to separate highly entangled components in X-ray data even in high-contrast scenarios, and can extract the spectrum and map of each physical component with high accuracy. A modification of the algorithm is proposed in order to improve the spectral fidelity in the case of strongly overlapping spatial components, and we investigate a resampling method to derive realistic uncertainties associated to the results of the algorithm. Applying the modified algorithm to the deep Chandra observations of Cassiopeia A, we are able to produce detailed maps of the synchrotron emission at low energies (0.6–2.2 keV), and of the red- and blueshifted distributions of a number of elements including Si and Fe K.