Based on the thiol-ene radical addition reaction, the branched hydroxyl functionalized modified styrene-butadiene-styrene triblock copolymer (SBS-g-OH) was prepared. According to chemical structure characterization, the order of reactivity proved that 1,2-vinyls had a priority to react with 2-mercaptoethanol (MCH) at a low functionalization degree of SBS-g-OH. A novel reaction kinetics model was set up to describe the addition reaction between MCH and 1,2-vinyls; then the effect of reaction time, temperature, and initial concentration ratio of MCH to 1,2-vinyls on functionalization degree of SBS-g-OH were investigated. Subsequently the performances of modified asphalt were investigated, where the softening point, ductility, and needle penetration of SBS-g-OH modified asphalt were almost equal to that of SBS modified asphalt, while the dispersibility of polymer in asphalt and storage stability of modified asphalt had been improved obviously using SBS-g-OH with an appropriate functionalization degree.