The stacking and burning of waste tires have posed a serious challenge to the natural environment. To make adequate use of this misplaced waste resource, the study prepared waste rubber powder (WR) and SBS compound-modified asphalt (WSMA) by applying the appropriate process. Subsequently, the low-temperature and rheological properties of WSMA were studied according to the force-ductility tests, Brookfield viscosity tests, frequency sweep tests, and multiple stress creep recovery tests. The Refutas model, the modified Sigmoid model, and the five-parameter viscoelastic constitutive model were employed to quantitatively analyze the rheological properties of WSMA influenced by the temperature, loading frequency, loading time, and stress level. The results revealed that the low-temperature performance, high-temperature performance, temperature sensitivity, rutting resistance, creep recovery performance, and stress sensitivity of WSMA with high WR content outperformed. The addition of 20% WR reduced the viscoplastic damage deformation by 85.97%, and the residual viscoplastic damage deformation can also be decreased by 99.46%. Moreover, FTIR results showed that adding WR reduced the value of IAR, which was helpful in improving the elasticity recovery ability of WSMA. The study is important for environmental protection and the improvement of asphalt performance.