Variations in the technical properties of asphalt binder and the aggregate gradation play a critical role in enhancing the performance and durability of pavement structures. This study investigated the synergistic effects of crumb rubber (GTR) and Sasobit additive on the mechanical properties and long-term performance of rubberized hot mix asphalt (HMA). The primary objective was to improve creep resistance, rutting resistance, and fatigue life of asphalt mixtures. Comprehensive experimental tests, including rotational viscosity of binders, high- and low-temperature performance grading, multiple stress creep and recovery (MSCR), linear amplitude sweep (LAS), dynamic creep, and four-point bending tests, were conducted on both binders and asphalt mixtures. The results indicated that the simultaneous incorporation of GTR and Sasobit significantly enhanced creep resistance, fatigue life, and rutting resistance of SMA mixtures and modified binders. Using response surface methodology (RSM), the optimal combination was determined as 14.7% GTR, 1.86% Sasobit, a mixing temperature of 167.8 °C, and 6.90% binder content. This optimal mix not only improved creep resistance, fatigue life, and thermal performance but also reduced plastic deformation of the mixtures. In addition to enhancing technical properties, the reduced mixing temperature contributed to energy savings and lower environmental emissions. Finally, the RSM model accurately predicted and optimized the properties of modified asphalt mixtures, providing an effective and economical strategy for producing durable and environmentally friendly pavements. These findings offer new insights into the utilization of recycled materials and polymer additives for improving the quality and sustainability of asphalt pavements.