Shack-Hartmann wavefront sensors (SHWS) are generally used to measure the wavefront shape of light beams. Measurement accuracy and the sensitivity of these sensors are important factors for better wavefront sensing. In this paper, we demonstrate a new type of SHWS with better measurement accuracy than the regular SHWS. The lenslet array in the regular SHWS is replaced with an array of coded phase masks, and the principle of coded aperture correlation holography (COACH) is used for wavefront reconstruction. Sharper correlation peaks achieved by COACH improve the accuracy of the estimated local slopes of the measured wavefront and consequently improve the reconstruction accuracy of the overall wavefront. Experimental results confirm that the proposed method provides a lower mean square wavefront error by one order of magnitude in comparison to the regular SHWS.