Nowadays, carbon nanotubes are a widely available material, especially multiwall carbon nanotubes. In addition to many other applications, they find use in all kinds of sensors, like deformation, motion, tensile and pressure responsive elements. Numerous applications in sensors for volatile organic compounds (VOCs) are reported as well, nevertheless, they mainly suffer from low selectivity. Therefore, in this research, a sensor containing MWCNTs dispersed in a functional polymer matrix was prepared. As a polymer matrix, styreneisoprene-styrene elastomer was chosen. The standard sensing mechanism of entangled MWCNTs is based on the quality of the contacts (charge transfer) between the individual nanotubes. In the prepared nanocomposite, the mechanism is modified due to the presence of the otherwise non-conductive matrix. The changes of conductivity depend on the response of the percolating nanotube filler network to the swelling of the polymer matrix due to adsorption of VOCs. The tested gas substances have high values of diffusion coefficient for the polymer, so they have a quick response. Then, the selectivity is ensured by differences in solubility of the tested VOCs in the polymer. The effect was demonstrated for four VOCs differing by their affinity to the polymer matrix, namely, heptane, toluene, acetone, and ethanol.