The paper presents the results of laboratory investigation on asphalt binders relaxation at low temperature, carried out in a ductilometer using the tensile test with continuous force measurement. Polymer modified asphalt binder samples consisting of a 50/70 penetration grade bitumen mixed with a concentrate of styrene-butadiene-styrene (SBS) modified bitumen—a 160/220 penetration grade bitumen modified with a SBS copolymer in the amount of 9%—were tested. Therefore, polymer modified binders containing 3%, 4.5%, 6% and 7.5% SBS, respectively, were obtained and investigated. Tensile tests were performed at −16 °C on samples before aging and subjected to short-term aging (RTFOT). Test results in the form of relaxation curves have been mathematically described using a modified generalized Maxwell model. Based on the acquired results, it was shown that the increase of the SBS copolymer content in asphalt binder precipitates the relaxation process, while aging slows down this phenomenon. It has also been proven that with increased content of SBS elastomer in asphalt binder, the effect of short-term aging on binder’s stress relaxation ability at low temperatures is reduced.