This study examined the synergistic effects of Styrene–Butadiene–Styrene (SBS) polymer and nanoclay on asphalt concrete mixture performance through a systematic experimental program using 4.5% SBS with varying nanoclay concentrations (0–8%). Performance evaluation included Indirect Tensile Strength (ITS), Indirect Tensile Resilient Modulus (E_RI), and Hamburg Wheel Tracking Tests (HWTT), along with novel quantitative analysis of visco-plastic and moisture resistance indices. Results demonstrated that 4.5% SBS with 6% nanoclay (4.5S6N) yielded optimal performance, achieving 38% increase in dry ITS, 68% improvement in wet ITS, and enhanced moisture resistance with Tensile strength Ratio (TSR) improving from 79.53% to 97.14%. The E_RI value increased by 39%, while rutting resistance improved by 39.3%. At this optimal concentration, nanoclay’s uniform dispersion and layered silicate structure created an effective reinforcement network, enhancing stress distribution and interfacial bonding with the SBS polymer network and asphalt components. However, exceeding 6% nanoclay content led to performance deterioration due to particle agglomeration. These findings demonstrate that optimized SBS–nanoclay modification effectively addresses both mechanical and moisture-related performance requirements for modern pavement applications.