The use of bent asymmetric Laue crystals to sagittally focus high-energy synchrotron X-rays calls for an understanding of the mechanisms affecting X-ray diffraction by such crystals. The rocking-curve width, a measurable quantity directly related to the distortion of the lattice planes, is the necessary first step towards such an understanding. A model is formulated for assessing the rocking-curve widths of sagittally bent Laue crystals, considering the elastic anisotropy. A method for depth-resolved measurement of the rocking curves was also developed to verify the model. The model successfully explains the wide range of rocking-curve widths of a large number of reflections from silicon crystals with two different orientations.