With the growing interest in bituminous construction materials, desulfurized crumb rubber (CR)/styrene–butadiene–styrene (SBS) modified asphalts have been investigated by many researchers as low-cost environmental-friendly road construction materials. This study aimed to investigate the rheological properties of desulfurized CR/SBS composite modified asphalt within various temperature ranges. Bending beam rheometer (BBR), linear amplitude sweep (LAS), and multiple stress creep recovery (MSCR) tests were performed on conventional CR/SBS composite modified asphalt and five types of desulfurized CR/SBS modified asphalts. Meanwhile, Burgers’ model and the Kelvin–Voigt model were used to derive nonlinear viscoelastic parameters and analyze the viscoelastic mechanical behavior of the asphalts. The experimental results indicate that both the desulfurized CR/SBS composite modifier and force chemical reactor technique can enhance the crosslinking of CR and SBS copolymer, resulting in an improved high-, intermediate-, and low-temperature performance of desulfurized CR/SBS composite modified asphalt. Burgers’ model was found to be apposite in simulating the creep stages obtained from MSCR tests for CR/SBS composite modified asphalts. The superior high-temperature performance of desulfurized CR/SBS modified asphalt prepared with 4% SBS, 20% desulfurized rubber, and a force chemical reactor time of 45 min contributes to the good high-temperature elastic properties of the asphalt. Therefore, this combination is recommended as an optimal preparation process. In summary, the desulfurization of crumb rubber and using the force chemical reactor technique are beneficial to composite asphalt performance and can provide a new way of utilizing waste tire rubber.