With the continuing growth of waste sulfur production from the petroleum industry processes, its utilization for the production of useful, low-cost, and environmentally beneficial materials is of primary interest. Elemental sulfur has a significant and established history in the modification of bitumen binders, while the sulfur-containing high-molecular compounds are limited in this field. Herein, we report a novel possibility to utilize the sulfur/organic copolymers obtained via the inverse vulcanization process as modifiers for bitumen binders. Synthesis and thermal characterization (TGA-DSC) of polysulfides derived from elemental sulfur (S8) and unsaturated organic species (dicyclopentadiene, styrene, and limonene) have been carried out. The performance of modified bitumen binders has been studied by several mechanical measurements (softening point, ductility, penetration at 25 °C, frass breaking point, adhesion to glass and gravel) and compared to the unmodified bitumen from the perspective of normalized requirements concerning polymer-modified bitumen. The interaction of bitumen binder with sulfur/organic modifier has been studied by means of FTIR spectroscopy and DSC measurements. The impact of the modification on the performance properties of bitumen has been demonstrated. The bitumen binders modified with sulfur/organic copolymers are in general less sensitive to higher temperatures (higher softening point up to 7 °C), more resistant to permanent deformations (lower penetration depth), and more resistant to aging processes without intrusive deterioration of parameters at lower temperatures. What is more, the modification resulted in significantly higher adhesion of bitumen binders to both glass (from 25% up to 87%) and gravel surfaces in combination with a lower tendency to form permanent deformations (more elastic behavior of the modified materials).