Ghost imaging (GI) is a method to nonlocally image an object with a single-pixel detector. However, the speckle's transverse size at the object plane limits the system's imaging resolution for conventional GI linear reconstruction algorithm. By combining the sparsity constraint of imaging object with ghost imaging method, we demonstrate experimentally that ghost imaging via sparsity constraint (GISC) can dramatically enhance the imaging resolution even using the random measurements far below the Nyquist limit. The image reconstruction algorithm of GISC is based on compressive sensing. Factors affecting the reconstruction quality of high-resolution GISC, such as the receiving system's numerical aperture and the object's sparse representation basis, are also investigated experimentally. This high-resolution imaging technique will have great applications in the microscopy and remote-sensing areas.