Embedded-atom molecular dynamics simulations were used to follow the diffusion dynamics of compact Pt clusters with up to 19 atoms on Pt (111) surfaces. The results reveal a novel cluster diffusion mechanism, involving successive shear translations of adjacent subcluster regions, which give rise to reptation, a snake-like gliding motion. We show that for compact clusters with 4 to 6 atoms, this mechanism competes energetically with that of island diffusion through concerted motion. However, as the cluster size increases from >7 to similar or equal to 20 atoms, reptation becomes the energetically favored diffusion mechanism. The concerted shear motion of subcluster regions, leading to reptation, is also shown to play a significant role in dendritic-to-compact morphological transitions of Pt islands.