Aiming at fabricating high damping rubber composites, the acrylic rubber ACM was incorporated with sliding graft copolymer (SGC) materials. SGC is a novel supramolecular material with sliding crosslink junctions, and it acts as a high damping phase in ACM/SGC composites. Fourier transform infrared spectroscopy reveals the presence of two types of hydrogen bonds in ACM/SGC composites. Micro-structure analysis shows a clear sea-island phase structure. SGC particles disperse fairly uniformly in the ACM matrix. A wide interphase region exists between these two phases, indicating the good blend compatibility between ACM and SGC. The damping performance of ACM/SGC composites under dynamic shear strain and frequency condition significantly improved with the increase in SGC content. Specifically, the loss factor (tan δ) value of ACM/SGC (100/40) composite increased by 120% compared with that of neat ACM, according to the RPA results. The significantly improved damping property can be ascribed to the interfacial hydrogen bonds and the pulley effect of SGC molecules.