In this paper, ground tire rubber was modified with dicumyl peroxide and a variable content (in the range of 0–15 phr) of ethylene-vinyl acetate copolymers characterized by different vinyl acetate contents (in the range of 18–39 wt.%). Modification of ground tire rubber was performed via an auto-thermal extrusion process in which heat was generated during internal shearing of the material inside the extruder barrel. The processing, performance properties, and storage stability of modified reclaimed ground tire rubber were evaluated based on specific mechanical energy, infrared camera images, an oscillating disc rheometer, tensile tests, equilibrium swelling, gas chromatography combined with a flame ionization detector, and gas chromatography with mass spectrometry. It was found that the developed formulas of modified GTR allowed the preparation of materials characterized by tensile strengths in the range of 2.6–9.3 MPa and elongation at break in the range of 78–225%. Moreover, the prepared materials showed good storage stability for at least three months and satisfied processability with commercial rubbers (natural rubber, styrene-butadiene rubber).