Plastomer-modified asphalt binders meeting MSCR specifications, asphalt paving materials with such asphalt binders, and methods for fabricating such asphalt binders are provided. The asphalt binder contains a base asphalt and a plastomer. If the plastomer has a drop point no greater than about 139° C., the asphalt binder further contains sulfur; sulfur-containing compounds, such as hydrocarbyl polysulfides and thiuram disulfides; phenolic resins; metal oxides; or a combination thereof. The asphalt binder is substantially free of elastomer.