The running-in of cylinder liner-piston rings (CLPRs) is the most important process that must be performed before a marine diesel engine can be operated. The quality of running-in directly affects the reliability of a CLPR. The surface texture of a CLPR has been proven to significantly affect its lubrication performance. In this study, the tribological behavior of a CLPR during running-in is investigated. Three types of surface textures are generated on the CLPR via laser processing: dimple texture on piston rings, groove texture on cylinder liners, and co-texture on both sides. Subsequently, a series of tests are performed on a slice tester. A load of 300 N (1.64 MPa) is applied, and two speeds (50 and 100 rpm) are adopted. The CLPR running-in quality is characterized based on three parameters, i.e., the friction coefficient, contact resistance, and wear topography. Experimental results show that, compared with a non-textured surface, the three types of surface textures mentioned above improved the friction performance during running-in. The lubricant supply capacity of the dimple texture on the piston ring, as a mobile oil reservoir, is stronger than that of the groove texture on the cylinder liner serving as a static oil reservoir. By contrast, the wear resistance of the dimple texture, as a movable debris trap on the piston ring, is weaker than that of the groove texture on the cylinder liner, which serves as a static debris trap. It is demonstrated that the co-texture combines the advantages of dimples and groove textures. Compared with non-textured surfaces, the friction coefficient decreased the most at 100 rpm (44.5%), and the contact resistance improved the most at 50 rpm (352.9%). The coupling effect provides the surface with improved running-in quality by optimizing the tribological performance, particularly at the dead center. This study provides guidance for the tribological design and manufacturing of CLPR in marine diesel engines.