Nanocomposites, based on an ethylene vinyl acetate (EVA) copolymer with a vinyl acetate content of 34 wt % and varying amounts of multiwall carbon nanotubes (MWCNTs), were prepared by an electrospinning method. The dispersibility of the MWCNTs in the solution was improved by using cholesteryl 1-pyrenecarboxylate (PyChol) as a compatibilizer. The transmission electron microscopy images showed that the MWCNTs were aligned inside of the elastomeric matrix by the electrospinning process. The morphologies of the fibers were evaluated by scanning electron microscopy. When the amount of MWCNTs in the polymer solution reached 3 wt %, fibers with a diameter of 846 ± 447 nm were prepared. The chemical composition of the prepared fibers was investigated by Fourier transform infrared spectroscopy (FTIR) and X-ray photoelectron spectroscopy (XPS). FTIR results confirmed the presence of a carboxyl group, originating from the presence of PyChol. XPS results showed that the EVA fibers produced by electrospinning were oxidized in ethylene units, when comparing the spectra of the original EVA granules, but the presence of MWCNTs enhanced the stability of the EVA. The thermal stabilities of the fibers were tested with thermogravimetric analysis. The results confirmed that the presence of MWCNTs inside the fibers enhanced the thermal stabilities of the prepared nanocomposites.