Carbon nanotube (CNT)-based nanocomposites often possess properties such as high stiffness, electrical conductivity, and thermal stability and have been studied for various biomedical and biotechnological applications. However, the current design approaches utilize CNTs as physical fillers, and thus, the true potential of CNT-based nanocomposites has not been realized. Here, we introduce a general approach to fabricating stiff, elastomeric nanocomposites from poly(glycerol sebacate) (PGS) and CNTs. The covalent crosslinking between the nanotubes and polymer chains resulted in novel property combinations that are not observed in conventional nanocomposites. The addition of 1% CNTs resulted in a five-fold increase in the tensile modulus and a six-fold increase in compression modulus compared with PGS alone, which is far superior to the previously reported studies for CNT-based nanocomposites. Despite a significant increase in mechanical stiffness, the elasticity of the network was not compromised and the resulting nanocomposites showed more than 94% recovery. This study demonstrates that the chemical conjugation of CNTs to a PGS backbone results in stiff and elastomeric nanocomposites. Additionally, in vitro studies using human mesenchymal stem cells (hMSCs) indicated that the incorporation of CNTs into the PGS network significantly enhanced the differentiation potential of the seeded hMSCs, rendering them potentially suitable for applications ranging from scaffolding in musculoskeletal tissue engineering to biosensors in biomedical devices.