The conventional form of computed tomography using X-ray attenuation without any contrast agents is of limited use for the characterization of soft tissue in many fields of medical and biological studies. Grating-based phase-contrast computed tomography (gbPC-CT) is a promising alternative imaging method solving the low soft tissue contrast without the need of any contrast agent. While highly sensitive measurements are possible using conventional X-ray sources the spatial resolution does often not fulfill the requirements for specific imaging tasks, such as visualization of pathologies. The focus of this study is the increase in spatial resolution without loss of sensitivity. To overcome this limitation a super-resolution reconstruction based on sub-pixel shifts involving a deconvolution of the image data during each iteration is applied. In our study we achieve an effective pixel size of 28 μm with a conventional rotating anode tube and a photon-counting detector. We also demonstrate that the method can upgrade existing setups to measure tomographies with higher resolution. The results show the increase in resolution at high sensitivity and with the ability to make quantitative measurements. The combination of sparse sampling and statistical iterative reconstruction may be used to reduce the total measurement time. In conclusion, we present high-quality and high-resolution tomographic images of biological samples to demonstrate the experimental feasibility of super-resolution reconstruction.