We developed an off-axis quantitative phase microscopy that works for a light source with an extremely short spatial coherence length in order to reduce the diffraction noise and enhance the spatial resolution. A dynamic speckle wave whose coherence length is 440 nm was used as an illumination source. To implement an off-axis interferometry for a source of low spatial coherence, a diffraction grating was inserted in the reference beam path. In doing so, an oblique illumination was generated without rotation of the wavefront, which leads to a full-field and single-shot phase recording with improved phase sensitivity of more than a factor of 10 in comparison with coherent illumination. The spatial resolution, both laterally and axially, and the depth selectivity are significantly enhanced due to the wide angular spectrum of the speckle wave. We applied our method to image the dynamics of small intracellular particles in live biological cells. With enhanced phase sensitivity and speed, the proposed method will serve as a useful tool to study the dynamics of biological specimens.