We report high mechanical, dielectric, and thermal performance of carbon nanotubes (CNT) reinforced ethylene vinyl acetate (EVA) composites, fabricated using conventional melt extrusion processing. CNT have extremely high stiffness, electrical conductivity, and surface area, ensuring strong interactions with the polymer and effective reinforcement. The addition of CNT to EVA leads to an extremely high yield strength and Young’s modulus of the composites. The EVA composite produced, containing 5 wt% CNT, exhibited an almost 3-fold increase in Young’s modulus and a 2.2-fold improvement of yield strength compared to neat EVA. However, the composite maintained high deformation properties—a ductility of 1300%. Scanning electron microscopy analysis evidences the agglomeration of the CNT in the EVA/CNT composites. The EVA/CNT composites gained excellent electrostatic discharge properties—a surface resistivity in the range of 108 Ω/square. The observed thermal conductivity of the composites was increased by about 30% without losing the electrically insulating performance.