Novel uniaxial strain effect in the strongly correlated two-dimensional system beta'-(Cation)[Pd(dmit)(2)](2) [dmit 2-thioxo-1,3-dithiol-4,5-dithiolate] with the dimer-based crystal structure is described. When cation = Me4As, the strain along the crystallographic b axis readily suppresses the non-metallic behavior and induces a superconductivity at 4 K under 7 kbar. On the other hand, small strain around 2 kbar along both the a- and c-axis directions effectively enhances the non-metallic behavior. These unusual results suggest the critical role of intra- and inter-dimer interactions that affects the band width and the effective on-site Coulomb interaction.