The performance of SBS-modified asphalt mixtures can be enhanced by incorporating various types of fibers offering a cost-effective alternative to increasing the SBS content. In this study, three different fibers, Basalt, Polyester, and Lignin fibers were added to a 3% SBS-modified bitumen binder, and their performance was compared to a 7% SBS mixture without fibers. Laboratory tests, including indirect tensile strength and dynamic shear rheometer tests, were used to evaluate the mixtures. The indirect tensile strength of all samples was assessed at loading rates ranging from 10 to 70 MPa/s, while stiffness moduli were tested at frequencies of 5 Hz, 3.5 Hz, 1.9 Hz, and 1.2 Hz. Finite element simulations using the Burger’s Logit model have been performed and microstrain analysis has been carried out to assess rutting and fatigue damage, complementing the experimental results. The findings demonstrated that fiber-modified mixtures exhibited superior performance, with increased tensile strength and complex shear modulus. Among the fiber types, Basalt fiber showed the best results, outperforming the others, while Polyester and Lignin fibers displayed nearly identical performance. The Basalt fiber mixture outperformed the SBS-7% mixture by 25% in rutting resistance and 28% in fatigue damage.