To expand the application range of modified asphalt and mixtures and effectively reduce the aggregation of nanomaterials in asphalt, nanocarbon/styrene butadiene styrene (SBS)/rubber powder composite-modified asphalt is proposed. This paper presents a laboratory study on the performance of nanocarbon/copolymer SBS/rubber powder composite-modified asphalt, and nanocarbon particles modified by titanate coupling agents as modifiers are selected. The nanocarbon/copolymer SBS/rubber powder composite-modified asphalt was prepared by a high-speed shearing method. The physical properties and rheological performance were assessed using ductility tests, softening point tests, penetration tests, dynamic shear rheometer (DSR) tests, and bending beam rheometer (BBR) tests. Furthermore, the mixture properties, including the high-temperature stability, low-temperature cracking resistance, moisture stability, and freeze-thaw splitting, were evaluated in the laboratory. The micromorphology of the base asphalt and composite-modified asphalt was examined by scanning electron microscopy (SEM), and the reactions between the modifiers and AH-70 base asphalt were studied by Fourier transform infrared spectroscopy (FTIR). The results reveal that the surface-modified nanocarbon and rubber powder additives substantially increased the softening point and penetration index of the base asphalt, with little obvious influence on the low-temperature performance. In addition, when nanocarbon/copolymer SBS/rubber powder composite-modified asphalt was used, the high-temperature stability and low-temperature cracking resistance of the nanocarbon/copolymer SBS/rubber powder composite-modified asphalt mixture were approximately 1.3 times those of the nanocarbon/rubber powder asphalt mixture. In terms of the micromorphology and reaction, the addition of the nanocarbon can increase the compatibility between the base asphalt and rubber powder, and then the addition of copolymer SBS can improve the structure of nanocarbon (after surface modification)/rubber powder-modified asphalt to form a stable network. Moreover, the physical reaction plays the dominant role in the modification process for the rubber powder and base asphalt, and chemical reactions occur in the modification process for the surface-modified nanocarbon and base asphalt.