Poly(ethylene-co-vinyl acetate) (EVA)-based nanocomposites were prepared by melt mixing in an internal mixer with nanocalcium phosphate in different weight percentages. The nanocalcium phosphate with 10-nm size was prepared by the polymer-induced crystallization technique. The mechanical properties as well as the gas permeability tests were performed to analyze the effect of nanofiller incorporation in to the polymer. Molecular transport of different solvents such as water, benzene, and n-heptane was undertaken at room temperature for EVA nanocomposites with 0, 3, and 5% filler loading. Among the three, water showed less uptake and benzene showed maximum uptake. Transport parameters such as diffusion coefficient, sorption coefficient, and permeation coefficient were calculated, and all of them showed a decrease with respect to the filler loading. First-order kinetics model was applied to investigate the transport kinetics. Also, the sorption curves were compared to theoretical predictions and found to be in good agreement except for water.