A novel and efficient architecture of a structured-illumination digital holographic microscope (DHM) is presented. As the DHM operates at the diffraction limit, its spatial resolution on label-free imaging of transparent samples is improved by illuminating the sample with a structured illumination produced by a Fresnel's biprism. The theoretical analysis of the method forecasts a twofold improvement of the spatial resolution. The proposed method requires only two images to improve the spatial resolution, which eases the process of unmixing the high-resolution components by means of an unknown phase-shift procedure. Numerical modeling and experimental results validate the theoretical findings.